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ELEMENTS
OF
NATURAL PHILOSOPHY.
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it
-ELEMENTS
NATURAL PHILOSOPHY
SIR WILLIAM THOMSON, LL.D, D.C.L., F.R.S.,
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW,
AND
PETER GUTHRIE TAIT, M.A.,
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH.
PART I.
_ SECOND EDITION,
.f^T% B A R iT
AT THE UNIVERSITY PRESS.
i8;9
\_All Rights reserved^
VI PREFACE.
of notes of a part of the Glasgow course, drawn up for
Sir W. Thomson by John Ferguson, Esq., and printed for
the use of his students.
"We have had considerable difficulty in compiling this
treatise from the larger work — arising from the necessity for
condensation to a degree almost incompatible with the design
to omit nothing of importance : and we feel that it would
have given us much less trouble and anxiety, and would
probably have ensured a better result, had we written the
volume anew without keeping the larger book constantly
before us. The sole justification of the course we have pur-
sued is that wherever, in the present volume, the student
may feel further information to be desirable, he will have
no difficulty in finding it in the corresponding pages of the
larger work.
" A great portion of the present volume has been in type
since the autumn of 1863, and has been printed for the use
of our classes each autumn since that date."
To this we would now only add that the whole has been
revised, and that we have endeavoured to simplify those
portions which we have found by experience to present
difficulties to our students.
The present edition has been carefully revised by Mr W.
BURNSIDE, of Pembroke College : and an Ltdex, of which we
have recognized the necessity, has been drawn up for us by
Mr Scott Lang.
W. THOMSON.
P. G. TAIT.
January^ 1879.
CONTENTS.
DIVISION I. Preliminary.
PAGE
Chap. I. Kinematics i
„ II. Dynamical Laws and Principles . . 56
„ in. Experience in
„ IV. Measures and Instruments . . . 122
DIVISION II. Abstract Dynamics.
., V. Introductory 136
,, VI. Statics of a Particle. Attraction . . 140
,, VII. Statics of Solids and Fluids . . . 199
Appendix 282
DIVISION I.
PRELIMINARY.
CHAPTER L— KINEMATICS.
The word Dynamometer occurring in the Index on p. 288 should
have been Ergometer, that being the term which we shall in
future use to denote this class of instruments.
considered without reference to the bodies moved, or to the forces
producing the motion, or to the forces called into action by the
motion, constitute the subject of a branch of Pure Mathematics,
which is called Kinematics, or, in its more practical branches,
Mechanism.
5. Observation and experiment have afforded us the means of
translating, as it were, from Kinematics into Dynamics, and vice versd.
This is merely mentioned now in order to show the necessity for,
and the value of, the preliminary matter we are about to introduce.
6. Thus it appears that there are many properties of motion,
displacement, and deformation, which may be considered altogether
independently of force, mass, chemical constitution, elasticity, tempe-
rature, magnetism, electricity ; and that the preliminary consideration
of such properties in the abstract is of very great use for Natural
T. I
OF THE
UNIVERSITY
DIVISION I.
PRELIMINARY.
CHAPTER L— KINEMATICS.
1. The science which investigates the action of Force is called, by
the most logical writers, Dynamics. It is commonly, but erroneously,
called Mechanics ; a term employed by Newton in its true sense,
the Science of Machines, and the art of making them.
2. Force is recognized as acting in two ways :
1° so as to compel rest or to prevent change of motion, and
2" so as to produce or to change motion.
Dynamics, therefore, is divided into two parts, which are conveniently
called Statics and Kinetics.
3. In Statics the action of force in maintaining rest, or preventing
change of motion, the 'balancing of forces,' or Equilibrium, is
investigated ; in Kinetics, the action of force in producing or in
changing motion.
4. In Kinetics it is not mere inotion which is investigated, but the
relation oi forces to motion. The circumstances of mere motion,
considered without reference to the bodies moved, or to the forces
producing the motion, or to the forces called into action by the
motion, constitute the subject of a branch of Pure Mathematics,
which is called Kinematics, or, in its more practical branches.
Mechanism.
5. Observation and experiment have afforded us the means of
translating, as it were, from Kinematics into Dynamics, and vice versd.
This is merely mentioned now in order to show the necessity for,
and the value of, the preliminary matter we are about to introduce.
6. Thus it appears that there are many properties of motion,
displacement, and deformation, which may be considered altogether
independendy of force, mass, chemical constitution, elasticity, tempe-
rature, magnetism, electricity ; and that the preliminary consideration
of such properties in the abstract is of very great use for Natural
T. I
2 PRELIMINARY.
Philosophy. We devote to it, accordingly, the whole of this chapter ;
which will form, as it were, the Geometry of the subject, embracing
what can be observed or concluded with regard to actual motions,
as long as the cause is not sought. In this category we shall first take
up the free motion of a point, then the motion of a point attached
to an inextensible cord, then the motions and displacements of rigid
systems — and finally, the deformations of solid and fluid masses.
7. When a point moves from one position to another it must
evidently describe a continuous line, which may be curved or straight,
or even made up of portions of curved and straight lines meeting
each other at any angles. If the motion be that of a material particle,
however, there can be no abrupt change of velocity, nor of direction
unless where the velocity is zero, since (as we shall afterwards see)
such would imply the action of an infinite force. It is useful to con-
sider at the outset various theorems connected with the geometrical
notion of the path described by a moving point ; and these we shall
now take up, deferring the consideration of Velocity to a future
section, as being more closely connected with physical ideas.
8. The direction of motion of a moving point is at each instant
the tangent drawn to its path, if the path be a curve ; or the path
itself if a straight line. This is evident from the definition of the
tangent to a curve.
9. If the path be not straight the direction of motion changes
from point to point, and the rate of this change, per unit of length
of the curve, is called the Curvature. To exemplify this, suppose
two tangents, PT, QU, drawn to a circle,
and radii OP^ OQ, to the points of contact.
The angle between the tangents is the
qj change of direction between P and Q,
and the rate of change is to be measured
by the relation between this angle and the
length of the circular arc PQ. Now, if $
be the angle, s the arc, and r the radius, we
see at once that (as the angle between the radii is equal to the
angle between the tangents, and as the measure of an angle is the
ratio of the arc to the radius, § 54) r9 = s, and therefore - =- is the
measure of the curvature. Hence the curvature of a circle is in-
versely as its radius, and is measured, in terms of the proper unit of
curvature, simply by the reciprocal of the radius.
10. Any small portion of a curve may be approximately taken
as a circular arc, the approximation being closer and closer to the
truth, as the assumed arc is smaller. The curvature at any point
is the reciprocal of the radius of this circle for a small arc on each
side of the point.
11. If all the points of the curve lie in one plane, it is called 2. plane
curve, and if it be made up of portions of straight or curved lines it
KINEMATICS. 3
is called 2. plafie polygon. If the line; do not lie in one plane, we have
in one case what is called a curve of double curvature, in the other
a gauche polygon. The term ' curve of double curvature ' is a very
bad one, and, though in very general use, is, we hope, not inera-
dicable. The fact is, that there are not two curvatures, but only
a curvature (as above defined) of which the plane is continuously
changing, or twisting, round the tangent line. The course of such
a curve is, in common language, well called ' tortuous ; ' and the mea-
sure of the corresponding property is conveniently called Tortuosity.
12. The nature of this will be best understood by considering the
curve as a polygon whose sides are indefinitely small. Any two
consecutive sides, of course, lie in a plane — and in that plane the
curvature is measured as above; but in a curve which is not plane
the third side of the polygon will not be in the same plane with the
first two, and therefore the new plane in which the curvature is to
be measured is different from the old one. The plane of the curva-
ture on each side of any point of a tortuous curve is sometimes called
the Osculating Plajie of the curve at that point. As two successive
positions of it contain the second side of the polygon above men-
tioned, it is evident that the osculating plane passes from one position
to the next by revolving about the tangent to the curve.
13. Thus, as we proceed along such a curve, the curvature in
general varies ; and, at the same time, the plane in which the cur-
vature lies is turning about the tangent to the curve. The rate of
torsion, or the tortuosity, is therefore to be measured by the rate at
which the osculating plane turns about the tangent, per unit length
of the curve. The simplest illustration of a tortuous curve is the
thread of a screw. Compare § 41 {d).
14. The Integral Curvature, or whole change of direction, of an arc
of a plane curve, is the angle through which the tangent has turned
as we pass from one extremity to the other. The average curvature
of any portion is its whole curvature divided by its length. Suppose
a line, drawn through any fixed point, to turn so as always to be
parallel to the direction of motion of a point describing the curve :
the angle through which this turns during the motion of the point
exhibits what we have defined as the integral curvature. In esti-
mating this, we must of course take the enlarged modern meaning
of an angle, including angles greater than two right angles, and also
negative angles. Thus the integral curvature of any closed curve or
broken line, whether everywhere concave to the interior or not, is four
right angles, provided it does not cut itself. That of a Lemniscate,
g is zero. That of the Epicyloid @ is eight right angles ; and
so on.
15. The definition in last section may evidently be extended to
a plane polygon, and the integral change of direction, or the angle
between the first and last sides, is then the sum of its exterior angles,
all the sides being produced each in the direction in which the
I — 2
PRELIMINARY,
moving point describes it while passing round the figure. This is
true whether the polygon be closed or not. If closed, then, as long
as it is not crossed, this sunri is four right angles, — an extension of
the result in Euclid, where all reentrant polygons are excluded. In
the star- shaped figure ^^ , it is ten right angles, wanting the suna of
the five acute angles of the figure j i. e. it is eight right angles.
16. A chain, cord, or fine wire, or a fine fibre, filament, or hair,
may suggest, what is not to be found among natural or artificial pro-
ductions, a perfectly fiexible and i?iextensible line. The elementary
kinematics of this subject require no investigation. The mathematical
condition to be expressed in any case of it is simply that the distance
measured along the line from any one point to any other, remains
constant, however the line be bent.
17. The use of a cord in mechanism presents us with many
practical applications of this theory, which are in general extremely
simple; although curious, and not always very easy, geometrical
problems occur in connexion with it. We shall say nothing here
about such cases as knots, knitting, weaving, etc., as being exces-
sively difficult in their general development, and too simple in the
ordinary cases to require explanation.
18. The simplest and most useful applications are to the Pulley
and its combinations. In theory a pulley is simply a smooth body
which changes the dh'ectmz of a flexible and inextensible
cord stretched across part of its surface ; in practice (to
escape as much as possible of the inevitable friction)
it is a wheel, on part of whose circumference the cord
is wrapped.
(i) Suppose we have a single pulley B^ about which
the flexible and inextensible cord ABP is wrapped, and
suppose its free portions to be parallel.
If {A being fixed) a point P of the cord i'P'
be moved to P' , it is evident that each
of the portions AB and PB will be • ^
shortened by one-half of PP". Hence, i_j^'<
when P moves through any space in ■
the direction of the cord, the pulley B
moves in the same direction, through
half the space.
(2) If there be two cords and two pulleys, the
ends AA' being fixed, and the other end of AB
being attached to the pulley ^'— then, if all free
parts of the cord are parallel, when P is moved to
P', B' moves in the same direction through half the
space, and carries with it one end of the cord AB.
Hence B moves through half the space B did, that
is, one fourth of PP',
rP'
<^
B
K^'
KINEMATICS, 5
(3) And so on for any number of pulleys, if they be arranged
in the above manner. Similar considerations enable us to deter-
mine the relative motions of all parts of other systems of pulleys and
cords as long as all the free parts of the cords are parallel.
Of course, if a pulley \>Q.jixed^ the motion of a point of one end of
the cord to or fro77i it involves an equal motion of the other end
from or to it.
If the strings be not parallel, the relations of a single pulley or
of a system of pulleys are a little complex, but present no difficulty.
19. In the mechanical tracing of curves, a flexible and inextensible
cord is often supposed. Thus, in drawing an ellipse, the focal pro-
perty of the curve shows us that if we fix the ends of such a cord
to the foci and keep it stretched by a pencil, the pencil will trace
the curve.
By a ruler moveable about one focus, and a string attached to a
point in the ruler and to the other focus, and kept tight by a pencil
sliding along the edge of the ruler, the hyperbola may be described
by the help of its analogous focal property ; and so on.
20. But the consideration of evolutes is of some importance in
Natural Philosophy, especially in certain mechanical and optical
questions, and we shall therefore devote a section or two to this
application of Kinematics.
Def. If a flexible and inextensible string be fixed at one point
of a plane curve, and stretched along the curve, and be then
unwound in the plane of the curve, every point of it will describe
an Involute of the curve. The original curve is called the Evolute of
any one of the others.
21. It will be observed that we speak of an involute, and of the
evolute, of a curve. In fact, as will be easily seen, a curve can have
but one evolute, but it has an infinite number of involutes. For all
that we have to do to vary an involute, is to change the point of
the curve from which the tracing-point starts, or consider the invo-
lutes described by different points of the string ; and these will, in
general, be different curves. But the following section shows that
there is but one evolute.
22. Let AB be any curve, PQ a portion of an involute, pP^ qQ
positions of the free part of the string. It will be seen at once
that these must be tangents to the arc
AB at / and q. Also the string at
any stage, as pP, ultimately revolves
about p. Hence pP is normal (or per-
pendicular to the tangent) to the curve
PQ. And thus the evolute of PQ is y£
a definite curve, viz. the envelop of (or
line which is touched by) the normals drawn at every point of PQ,
or, which is the same thing, the locus of the centres of the circles
which have at each point the same tangent and curvature as the
curve PQ. And we may merely mention, as an obvious result of the
6 PRELIMINARY.
mode of tracing, that the arc qp is equal to the difference oi qQ and
pP, or that the 2x0, p A is equal to pP. Compare § 104.
23. The rate of motion of a point, or its rate of change of position^
is called its Velocity. It is greater or less as the space passed over
in a given time is greater or less : and it may be uniform^ i. e. the
same at every instant ; or it may be variable.
Uniform velocity is measured by the space passed over in unit of
time, and is, in general, expressed in feet or in metres per second ;
if very great, as in the case of light, it may be measured in miles per
second. It is to be observed that • Time is here used in the abstract
sense of a uniformly-increasing quantity — what in the differential cal-
culus is called an independent variable. Its physical definition is
given in the next chapter.
24. Thus a point, which moves uniformly with velocity v^ describes
a space of v feet each second, and therefore vt feet in / seconds,
t being any number whatever. Putting s for the space described
in / seconds, we have s = vt.
Thus with unit velocity a point describes unit of space in unit of
time.
25. It is well to observe here, that since, by our formula, we have
generally s
and since nothing has been said as to the magnitudes of s and /, we
may take these as small as we choose. Thus we get the same result
whether we derive v from the space described in a million seconds^ or
from that described in a millio?ith of a seco?id. This idea is very useful,
as it makes our results intelligible when a variable velocity has to be
measured, and we find ourselves obliged to approximate to its value
(as in § 28) by considering the space described in an interval so
short, that during its lapse the velocity does not sensibly alter in value.
26. When the point does not move uniformly, the velocity is
variable, or different at different successive instants : but we define
the average velocity during any time as the space described in that
time, divided by the time ; and, the less the interval is, the more
nearly does the average velocity coincide with the actual velocity at
any instant of the interval. Or again, we define the exact velocity at
any instant as the space which the point would have described in one
second, if for such a period it kept its velocity unchanged.
27. That there is at every instant a definite velocity for any moving
point, is evident to all, and is matter of everyday conversation. Thus,
a railway train, after starting, gradually increases its speed, and every
one understands what is meant by saying that at a particular instant it
moves at the rate of ten or of fifty miles an hour, — although, in the
course of an hour, it may not have moved a mile altogether. We
may suppose that, at any instant during the motion, the steam is so
adjusted as to keep the train running for some time at a uniform
velocity. This is the velocity which the train had at the instant in
KINEMATICS. 7
question. Without supposing any such definite adjustment of the
driving-power to be made, we can evidently obtain an approximation
to the velocity at a particular instant, by considering (§ 25) the
motion for so short a time, that during that time the actual variation
of speed may be small enough to be neglected.
28. In fact, if v be the velocity at either beginning or end, or at
any. instant, of an interval /, and s the space actually described in
that interval; the equation v = - (which expresses the definition of
the average velocity, § 26) is more and more nearly true, as the
velocity is more nearly uniform during the interval /; so that if we
take the interval small enough the equation may be made as nearly
exact as we choose. Thus the set of values —
Space described in one second.
Ten times the space described in the first tenth of a second,
A hundred „ „ „ hundredth „
and so on, give nearer and nearer approximations to the velocity at
the beginning of the first second.
The whole foundation of Newton's differential calculus is, in fact,
contained in the simple question, 'What is the rate at which the
space described by a moving point increases?' i.e. What is the
velocity of the moving point? Newton's notation for the velocity,
i. e. the rate at which s increases, or the Jluxion of J", is s. This
notation is very convenient, as it saves the introduction of a second
letter.
29. The preceding definition of velocity is equally applicable
whether the point move in a straight or a curved line; but, since,
in the latter case, the direction of motion continually changes, the
mere amount of the velocity is not sufficient completely to describe
the motion, and we must have in every such case additional data
to thoroughly specify the motion.
In such cases as this the method most commonly employed,
whether we deal with velocities, or (as we shall do farther on) with
accelerations and forces, consists in studying, not the velocity, accele-
ration, or force, directly, but its resolved parts parallel to any three
assumed directions at right angles to each other. Thus, for a train
moving up an incline in a N.E. direction, we may have the whole
velocity and the steepness of the incline given ; or we may express
the same ideas thus — the train is moving simultaneously northward,
eastward, and upward — and the motion as to amount and direction
will be completely known if we know separately the northward, east-
ward, and upward velocities — these being called the components of
the whole velocity in the three mutually perpendicular directions
N., E., and up.
30. A velocity in any direction may be resolved in, and perpen-
dicular to, any other direction. The first component is found by
multiplying the velocity by the cosine of the angle between the two
8 PRELIMINARY.
directions ; the second by using as factor the sine of the same angle.
Thus a point moving with velocity V up an Inclined Plane^ making
an angle a with the horizon, has a vertical velocity Fsin^; and a
horizontal velocity Fcos a.
Or it may be resolved into components in any three rectangular
directions, each component being found by multiplying the whole
velocity by the cosine of the angle between its direction and that of
the component. The velocity resolved in any direction is the sum
of the resolved parts (in that direction) of the three rectangular com-
ponents of the whole velocity. And if we consider motion in one
plane, this is still true, only we have but two rectangular com-
ponents.
31. These propositions are virtually equivalent to the following
obvious geometrical construction : —
To compound any two velocities as OA^ OB in the figure ; where
^ ^ OA, for instance, represents in magni-
/- "J^^ ^^^^ ^"^ direction the space which
/ ^^■'</ would be described in one second by
/ ^^^ y/ a point moving with the first of the
/ ^^^ / given velocities — and similarly OB for
Z,.-^ / the second ; from A draw A C parallel
^^_ / and equal to OB. Join OC-. then OC
^ -A. is the resultant velocity in magnitude
and direction.
OC is evidently the diagonal of the parallelogram two of whose
sides are OA, OB.
Hence the resultant of any two velocities as OA, AC, in the
figure, is a velocity represented by the third side, OC, of the triangle
OAC.
Hence if a point have, at the same time, velocities represented by
OA, A C, and CO, the sides of a triangle taken ifi the same order, it
is at rest.
Hence the resultant of velocities represented by the sides of any
closed polygon whatever, whether in one plane or not, taken all in
the same order, is zero.
Hence also the resultant of velocities represented by all the sides
of a polygon but one, taken in order, is represented by that one
taken in the opposite direction.
When there are two velocities, or three velocities, in two or in
three rectangular directions, the resultant is the square root of the
sum of their squares ; and the cosines of its inclination to the given
directions are the ratios of the components to the resultant.
32. The velocity of a point is said to be accelerated or retarded
according as it increases or diminishes, but the word acceleration is
generally used in either sense, on the understanding that we may
regard its quantity as either positive or negative : and (§ 34) is
farther generalized so as to include change of direction as well as
change of speed. Acceleration of velocity may of course be either
KINEMATICS. 9
uniform or variable. It is said to be uniform when the point receives
equal increments of velocity in equal times, and is then measured by
the actual increase of velocity per unit of time. If we choose as the
unit of acceleration that which adds a unit of velocity per unit of
time to the velocity of a point, an acceleration measured by a will
add a units of velocity in unit of time — and, therefore, a t units of
velocity in / units of time. Hence if v be the change in the velocity
during the interval /,
V = at, or a =^ J.
33. Acceleration is variable when the point's velocity does not
receive equal increments in successive equal periods of time. It is
then measured by the increment of velocity, which would have been
generated in a unit of time had the acceleration remained throughout
that unit the same as at its commencement. The average accelera-
tion during any time is the whole velocity gained- during that time,
divided by the time. In Newton's notation v is used to express the
acceleration in the direction of motion ; and, \i v = s as in § 28, we
have a = v = 's.
34. But there is another form in which acceleration may manifest
itself. Even if a point's velocity remain unchanged, yet if its direc-
tion of motion change, the resolved parts of its velocity in fixed
directions will, in general, be accelerated.
Since acceleration is merely a change of the component velocity
in a stated direction, it is evident that the laws of composition and
resolution of accelerations are the same as those of velocities.
We therefore expand the definition just given, thus : — Acceleration
is the rate of change of velocity ivhether that change take place in the
directiofi of motion or not,
35. What is meant by change of velocity is evident from § 31.
For if a velocity OA become OC^ its change is AC, or OB.
Hence, just as the direction of motion of a point is the tangent to
its path, so the direction of acceleration of a moving
point is to be found by the following construction : —
From any point O draw lines OP, OQ, etc., repre-
senting in magnitude and direction the velocity of the
moving point at every instant. (Compare § 49.) The
points, F, Q, etc., must form a continuous curve, for
(§ 7) OF cannot change abruptly in direction. Now
if ^ be a point near to F, OF and OQ represent two
successive values of the velocity. Hence FQ is the
whole change of velocity during the interval. As the O
interval becomes smaller, the direction FQ more and more nearly
becomes the tangent at F. Hence the direction of acceleration is
that of the tangent to the curve thus described.
The magnitude of the acceleration is the rate of change of velocity,
and is therefore measured by the velocity of F in the curve FQ.
lo PRELIMINARY,
36. Let a point describe a circle, ABD^ radius R^ with uniform
velocity V, Then, to determine the direction of acceleration, we
must draw, as below, from a fixed point (9, lines OP, OQ, etc.,
representing the velocity at A, B, etc., in direction and magnitude.
Since the velocity in ABD is constant, all the lines OP, OQ, etc.,
will be equal (to F), and there-
fore PQS is a circle whose
centre is O. The direction of
acceleration at A is parallel to
the tangent at P, that is, is per-
pendicular to OP, i.e. to Aa,
and is therefore that of the
radius AC.
Now P describes the circle
PQS, while A describes ABD.
Hence the velocity of P is to
that of ^ as OP to CA, i.e. as Fto R; and is therefore equal to
r'^'^'r^
and this (§.35) is the amount of the acceleration in the circular path
ABI?.
37. The whole acceleration in any direction is the sum of the
components (in that direction) of the accelerations parallel to any
three rectangular axes — each component acceleration being found
by the same rule as component velocities (§ 34), that is, by multiply-
ing by the cosine of the angle between the direction of the accelera-
tion and the line along which it is to be resolved.
38. When a point moves in a curve the whole acceleration may
be resolved into two parts, one in the direction of the motion and
equal to the acceleration of the velocity; the other towards the
centre of curvature (perpendicular therefore to the direction of mo-
tion), whose magnitude is proportional to the square of the velocity
and also to the curvature of the path. The former of these changes
the velocity, the other affects only the form of the path, or the
direction of motion. Hence if a moving point be subject to an
acceleration, constant or not, whose direction is continually perpen-
dicular to the direction of motion, the velocity will not be altered —
and the only effect of the acceleration will be to make the point
move in a curve whose curvature is proportional to the acceleration
at each instant, and inversely as the square of the velocity.
39. In other words, if a point move in a curve, whether with a
uniform or a varying velocity, its change of direction is to be re-
garded as constituting an acceleration towards the centre of curva-
ture, equal in amount to the square of the velocity divided by the
radius of curvature. The whole acceleration will, in every case, be
the resultant of the acceleration thus measuring change of direction
and the acceleration of actual velocity along the curve.
KINEMATICS. ii
40. If for any case of motion of a point we have given the whole
velocity and its direction, or simply the components of the velocity
in three rectangular directions, at any tb7ie^ or, as is most commonly
the case, for any position ; the determination of the form of the path
described, and of other circumstances of the motion, is a question of
pure mathematics, and in all cases is capable (if not of an exact
solution, at all events) of a solution to any degree of approximation
that n^y be desired.
This is true also if the total acceleration and its direction at every
instant, or simply its rectangular components, be given, provided the
velocity and its direction, as well as the position of the point, at any
one instant be given. But these are, in general, questions requiring
for their solution a knowledge of the integral calculus.
41. From the principles already laid down, a great many interest-
ing results may be deduced, of which we enunciate a few of the
simpler and more important.
{a) If the velocity of a moving point be uniform, and if its direction
revolve uniformly in a plane, the path described is a circle.
(^) If a point moves in a plane, and its component velocity
parallel to each of two rectangular axes is proportional to its dis-
tance from that axis, the path is an ellipse or hyperbola whose
principal diameters coincide with those axes; and the acceleration
is directed to or from the centre of the curve at every instant
(§§ 66, 78).
{c) If the components of the velocity parallel to each axis be equi-
multiples of the distances from the other axis, the path is a straight
line passing through the origin.
{d) When the velocity is uniform, but in a direction revolving
uniformly in a right circular cone, the motion of the point is in a
circular helix whose axis is parallel to that of the cone.
42. When a point moves uniformly in a circle of radius R^ with
velocity F", the whole acceleration is directed towards the centre, and
has the constant value -^. See § 36.
43. With uniform acceleration in the direction of motion, a point
describes spaces proportional to the squares of the times elapsed
since the commencement of the motion. This is the case of a body
falling vertically in vacuo under the action of gravity.
In this case the space described in any interval is that which would
be described in the same time by a point moving uniformly with a
velocity equal to that at the middle of the interval. In other words,
the average velocity (when the acceleration is uniform) is, during any
interval, the arithmetical mean of the initial and final velocities. For,
since the velocity increases uniformly, its value at any time before the
middle of the interval is as much less than this mean as its value
at the same time after the middle of the interval is greater than the
PRELIMINARY.
mean : and hence Its value at the middle of the Interval must be the
mean of its first and last values.
In symbols ; if at time / = o the velocity was F, then at time / it is
v= F+af.
Also the space (x) described is equal to the product of the time by the
average velocity. But we have just shown that the average velocity is
and therefore x = Vt + \at^.
Hence, by algebra,
V +2ax = V'+2 Vat + a'f - ( V-\- atf = v\
or 1 z;- - 1 F^ = ax.
If there be no initial velocity our equations become
v-at^ x-\af, \v^ = ax.
Of course the preceding formulae apply to a constant retardation, as
in the case of a projectile moving vertically upwards, by simply giving
a a negative sign.
44. When there Is uniform acceleration in a constant direction,
the path described is a parabola, whose axis is parallel to that
direction. This is the case of a projectile moving in vacuo.
For the velocity ( V) in the original direction of motion remains
unchanged ; and therefore, in time /, a space Vt is described parallel
to this line. But in the same Interval, by the above reasoning, we see
that a space \af is described parallel to the direction of acceleration.
Hence, if AP be the direction of motion at A^ AB the direction
of acceleration, and Q the position of the point at time t\
draw QP parallel to BA, meeting AP m
P\ then
AP=Vt, PQ^\at\
Hence
AP' = ^^PQ.
This is a property of a parabola, of which
the axis is parallel to AB; AB being a
diameter, and AP a tangent. ' If (9 be
the focus of this curve, we know that
AP' = ^OA.PQ.
Hence
0A=—,
2a
B
Also OA Is known In direction, for AP
between the focal distance of a point and
and is therefore known
bisects the angle, OAC,
the diameter through it.
45. When the acceleration, whatever (and however varying) be
its magnitude, is directed to a fixed point, the path is in a plane
KINEMATICS.
13
passing through that point ; and in this plane the areas traced out by
the radius-vector are proportional to the times employed.
Evidently there is no acceleration perpendicular to the plane con-
taining the fixed point and the line of motion of the moving point
at any instant; and there being no velocity perpendicular to this
plane at starting, there is therefore none throughout the motion;
thus the point moves in the plane. For the proof of the second
part of th^ proposition we must make a slight digression.
46. The Moment of a velocity or of a force about any point is
the product of its magnitude into the perpendicular from the point
upon its direction. The moment of the resultant velocity of a par-
ticle about any point in the plane of the components is equal to the
algebraic sum of the moments of the components, the proper sign of
each moment depending on the direction of motion about the point.
The same is true of moments of forces and of moments of momentum,
as defined in Chapter II. ^
First, consider two component motions, AB and A C, and let AD
be their resultant (§ 31). Their half-moments round the point O
are respectively the areas OAB, OCA. Now OCA, together with
half the area of the parallelogram CABB, is equal to OBD. Hence
the sum of the two half-moments together ^
with half the area of the parallelogram is .^^..^
equal to A OB together with BOD, that is /jW
to say, to the area of the whole figure / j \ \.
OABD. But ABD, a part of this figure,
is equal to half the area of the parallelo-
gram; and therefore the remainder, OAD,
is equal to the sum of the two half-mo-
ments. But OAD is half the moment of the •4. B
resultant velocity round the point O. Hence the moment of the
resultant is equal to the sum of the moments of the two components.
By attending to the signs of the mom.ents, we see that the proposi-
tion holds when O is within the angle CAB.
If there be any number of component rectilineal motions, we may
compound them in order, any two taken together first, then a third,
and so on; and it follows that the sum of their moments is equal to
the moment of their resultant. It follows, of course, that the sum of
the moments of any number of component velocities, all in one plane,
into which the velocity of any point may be resolved, is equal to the
moment of their resultant, round any point in their plane. It follows
also, that if velocities, in different directions all in one plane, be suc-
cessively given to a moving point, so that at any time its velocity is
their resultant, the moment of its velocity at any time is the sum of the
moments of all the velocities which have been successively given to it.
47. Thus if one of the components always passes through the
point, its moment vanishes. This is the case of a motion in which
the acceleration is directed to a fixed point, and we thus prove the
second theorem of § 45, that in the case supposed the areas described
14 PRELIMINARY.
by the radius-vector are proportional to the times; for, as we have
seen, the moment of the velocity is double the area traced out by the
radius-vector in unit of time.
48. Hence in this case the velocity at any point is inversely as
the perpendicular from the fixed point to the tangent to the path or
the momentary direction of motion.
For the product of this perpendicular and the velocity at any
instant gives double the area described in one second about the
fixed point, which has just been shown to be a constant quantity.
As the kinematical propositions with which we are dealing have
important bearings on Physical Astronomy, we enunciate here Kepler's
Laws of Planetary Motion. They were deduced originally from
observation alone, but Newton explained them on physical principles
and showed that they are applicable to comets as well as to planets.
I. Each planet describes an Ellipse [with comets this may be any
Conic Section] of which the Sun occupies one focus.
II. The radius-vector of each planet describes equal areas in
equal times.
III. The square of the periodic time [in an elliptic orbit] is pro-
portional to the cube of the major axis.
Sections 45 — 47, taken in connexion with the second of these
laws, show that the acceleration of the motion of a planet or comet
is along the radius-vector.
49. If, as in § 35, from any fixed point, lines be drawn at every
instant representing in magnitude and direction the velocity of a point
describing any path in any manner, the extremities of these lines
form a curve which is called the Hodograph. The fixed point from
which these lines are drawn is called the hodographic origin. The
invention of this construction is due to Sir W. R. Hamilton; and
one of the most beautiful of the many remarkable theorems to which
it leads is this : The Hodograph for the motio?i of a planet or comet is
always a circle^ whatever be the form and dimensions of the orbit. The
proof will be given immediately.
It was shown (§35) that an arc of the hodograph represents the
change of velocity of the moving point during the corresponding
time ; and also that the tangent to the hodograph is parallel to the
direction, and the velocity in the hodograph is equal to the amount
of the acceleration of the moving point.
When the hodograph and its origin, and the velocity along it, or
the time corresponding to each point of it, are given, the orbit may
easily be shown to be determinate.
[An important improvement in nautical charts has been suggested
by Archibald Smiths It consists in drawing a curve, which may
be called the tidal hodograph with reference to any point of a chart
for which the tidal currents are to be specified throughout the chief
tidal period (twelve lunar hours). Numbers from I. to XII. are placed
at marked points along the curve, corresponding to the lunar hours.
* Proc. R. S. 1865.
KINEMATICS,
15
— B
Smith's curve is precisely the Hamiltonian hodograph for an imagi-
nary particle moving at each instant with the same velocity and the
same direction as the particle of fluid passing, at the same instant,
through the point referred to.]
50. In the case of a projectile (§ 44), the horizontal component
of the velocity is unchanged, and the vertical component increases
uniformly. Hence the hodograph is a vertical straight line, whose
distance from the origin is the horizontal velocity, and which is
described urtiformly.
51. To prove Hamilton's proposition (§ 49), let APB be a portion
of a conic section and S one focus. Let P move so that SP
describes equal areas in equal times, that is (§ 48),
let the velocity be inversely as the perpendicular
SY from S to the tangent to the orbit. \i ABP
be an ellipse or hyperbola, the intersection of the
perpendicular with the tangent lies in the circle
YAZ, whose diameter is the major axis. Produce
YS to cut the circle again in Z Then YS. SZ is
constant, and therefore SZ is inversely as SY, that
is, SZ is proportional to the velocity at P. Also
SZ is perpendicular to the direction of motion PY,
and thus the circular locus of Z is the hodograph turned through a
right angle about S in the plane of the orbit. If APB be a parabola,
^ F is a straight line. But if another point UhQ taken in YS pro-
duced, so that YS. SU"\s constant, the locus of 6^is easily seen to be a
circle. Hence the proposition is generally true for all conic sections.
The hodograph surrounds its origin [as the figure shows] if the orbit
be an ellipse, passes through it when the orbit is a parabola, and the
origin is without the hodograph if the orbit is a hyperbola.
52. A reversal of the demonstration of § 5 1 shows that, if the
acceleration be towards a fixed point, and if the hodograph be a
circle, the orbit must be a conic section of which the fixed point
is a focus.
But we may also prove this important proposition as follows:
Let A be the centre of the circle, and O the hodographic origin.
Join OA and draw the perpendiculars
PM to OA and ON to PA. Then OP
is the velocity in the orbit : and ON, being
parallel to the tangent at P, is the direc-
tion of acceleration in the orbit; and is
therefore parallel to the radius-vector to
the fixed point about which there is equable
description of areas. The velocity parallel
to the radius-vector is therefore ON, and
the velocity perpendicular to the fixed line
OA is PM.
^ , ON OA
^'''-PM=AP
constant.
i6 PRELIMINARY.
Hence, in the orbit, the velocity along the radius-vector is pro-
portional to that perpendicular to a fixed line : and therefore the
radius-vector of any point is proportional to the distance of that
point from a fixed line — a property belonging exclusively to the
conic sections referred to their focus and directrix.
53. The path which, in consequence of Aberration^ a fixed star
seems to describe, is the hodograph of the earth's orbit, and is
therefore a circle whose plane is parallel to the plane of the
ecliptic.
54. When a point moves in any manner, the line joining it with
a fixed point generally changes its direction. If, for simplicity, we
consider the motion to be confined to a plane passing through the
fixed point, the angle which the joining line makes with a fixed line
in the plane is continually altering, and its rate of alteration at any
instant is called the Angular Velocity of the first point about the
second. If uniform, it is of course measured by the angle described
in unit of time ; if variable, by the angle which would have been
described in unit of time if the angular velocity at the instant in
question were maintained constant for so long. In this respect the
process is precisely similar to that which we have already explained
for the measurement of velocity and acceleration.
We may also speak of the angular velocity of a moving plane
with respect to a fixed one, as the rate of increase of the angle
contained by them ; but unless their line of intersection remain fixed,
or at all events parallel to itself, a somewhat more laboured statement
is required to give a complete specification of the motion.
55. The unit angular velocity is that of a point which describes,
or would describe, unit angle about a fixed point in unit of time. The
usual unit angle is (as explained in treatises on plane trigonometry) that
which subtends at the centre of a circle an arc whose length is equal
to the radius; being an angle of = S7"*29578 ... = 57"i7'44"-8
TT
nearly.
56. The angular velocity of a point in a plane is evidently to be
found by dividing the velocity perpendicular to the radius-vector by
the length of the radius-vector.
57. When the angular velocity is variable its rate of increase or
diminution is called the Angular Acceleration, and is measured with
reference to the same unit angle.
58. When one point describes uniformly a circle about another,
the time of describing a complete circumference being JJ we have
the angle 27r described uniformly in T\ and, therefore, the angular
velocity is -= . Even when the angular velocity is not uniform, as in
a planet's motion, it is useful to introduce the quantity -=, , which is
then called the mean angular velocity.
KINEMATICS. 17
59. When a point moves uniformly in a straight line its angular
velocity evidently diminishes as it recedes from the point about
which the angles are measured, and it may easily be shown that
it varies inversely as the square of the distance from this point.
The same proposition is true for miy path, when the acceleration is
towards the point about which the angles are measured : being
merely a different mode of stating the result of § 48.
60. The intensity of heat and light emanating from a point, or
from a uniformly radiating spherical surface, diminishes according to
the inverse square of the distance from the centre. Hence the rate
at which a planet receives heat and light from the sun varies in
simple proportion to the angular velocity of the radius-vector. Hence
the whole heat and light received by the planet in any time is pro-
portional to the whole angle turned through by its radius-vector in
the same time.
61. A further instance of this use of the idea of angular velocity
may now be given, to solve the problem of finding the hodograph
(§35) for any case of motion in which the acceleration is directed to
a fixed point, and varies inversely as the square of the distance from
that point. The velocity of P, in the hodograph PQ^ being the
acceleration in the orbit, varies inversely as the square of the
radius-vector; and therefore (§ 59) directly as the
angular velocity. Hence the arc of PQ^, described
in any time, is proportional to the corresponding
angle-vector in the orbit, i.e. to the angle through
which the tangent to PQ has turned. Hence (§ 9)
the curvature of PQ is constant, or PQ is a circle.
This demonstration, reversed, proves that if the
hodograph be a circle, and the acceleration be to-
wards a fixed point, the acceleratioti varies inversely
as the square of the distance of the moving point
from the fixed point.
62. From §§ 61, 52, it follows that when a particle moves with
acceleration towards a fixed point, varying inversely as the square
of the distance, its orbit is a conic section, with this point for one
focus. And conversely (§§ 47, 51, 52), if the orbit be a conic sec-
tion, the acceleration, if towards either focus, varies inversely as the
square of the distance : or, if a point moves in a conic section,
describing equal areas in equal times by a radius-vector through
a focus, the acceleration is always towards this focus, and varies
inversely as the square of the distance. Compare this with the first
and second of Kepler's Laws, § 48.
63. All motion that we are, or can be, acquainted with, is Relative
merely. We can calculate from astronomical data for any instant
the direction in which, and the velocity with which, we are moving
on account of the earth's diurnal rotation. We may compound this
with the (equally calculable) velocity of the earth in its orbit. This
resultant again we may compound with the (roughly-known) velocity
r
^\ r? {^ A n y '%,
OF THE ^
f UNIVERSITY I
1 8 PRELIMINARY.
of the sun relatively to the so-called fixed stars ; but, even if all
these elements were accurately known, it could not be said that we
had attained any idea of an absolute velocity ; for it is only the sun's
relative motion among the stars that we can observe; and, in all
probability, sun and stars are moving on (it may be with incon-
ceivable rapidity) relatively to other bodies in space. We must there-
fore consider how, from the actual motions of a set of bodies, we
may find their relative motions with regard to any one of them ; and
how, having given the relative motions of all but one with regard to
the latter, and the actual motion of the latter, we may find the actual
motions of all. The question is very easily answered. Consider
for a moment a number of passengers walking on the deck of a
steamer. Their relative motions with regard to the deck are what we
immediately observe, but if we compound with these the velocity of
the steamer itself we get evidently their actual motion relatively to
the earth. Again, in order to get the relative motion of all with
regard to the deck, we eliminate the motion of the steamer alto-
gether ; that is, we alter the velocity of each relatively to the earth
by compounding with it the actual velocity of the vessel taken in
a reversed direction.
Hence to find the relative motions of any set of bodies with regard
to one of their number, imagine, impressed upon each in composition
with its own motion, a motion equal and opposite to the motion of
that one, which will thus be reduced to rest, while the motions of the
others will remain the same with regard to it as before.
Thus, to take a very simple example, two trains are running in
opposite directions, say north and south, one with a velocity of fifty,
the other of thirty, miles an hour. The relative velocity of the second
with regard to the first is to be found by imagining impressed on
both a southward velocity of fifty miles an hour ; the effect of this
being to bring the first to rest, and to give the second a southward
velocity of eighty miles an hour, which is the required relative
motion.
Or, given one train moving north at the rate of thirty miles an
hour, and another moving west at the rate of forty miles an hour.
The motion of the second relatively to the first is at the rate of fifty
miles an hour, in a south-westerly direction inclined to the due west
direction at an angle of tan~^|. It is needless to multiply such
examples, as they must occur to every one.
64. Exactly the same remarks apply to relative as compared with
absolute acceleration, as indeed we may see at once, since accelera-
tions are in all cases resolved and compounded by the same law as
velocities.
65. The following proposition in relative motion is of consider-
able importance : —
Any two moving points describe similar paths relatively to each
other and relatively to any point which divides in a constant ratio
the line joining them.
KINEMATICS. 19
Let A and B be any simultaneous positions of the points. Take
G or G' in AB such that the ratio 7;^-^ or ^^^^ has a constant
value. Then, as the form of the relative 1 1 1 -+
path depends only upon the length and G A. G B
direction of the line joining the two points at any instant, it is obvious
that these will be the same for A with regard to B^ as for B with
regard to A^ saving only the inversion of the direction of the joining
line. Hence ^'s path about A is A''s> about B turned through two
right angles. And with regard to G and G' it is evident that the
directions remain the same, while the lengths are altered in a given
ratio ; but this is the definition of similar curves.
66. An excellent example of the transformation of relative into
absolute motion is afforded by the family of Cycloids. We shall in
a future section consider their mechanical description, by the rolling
of a circle on a fixed straight line or circle. In the meantime, we
take a different form of statement, which however leads to precisely
the same result.
The actual path of a point which revolves uniformly in a circle
about another point — the latter moving uniformly in a straight line
or circle in the same plane — belongs to the family of Cycloids.
67. As an additional illustration of this part of our subject, we
may define as follows :
If one point A executes any motion whatever with reference to
a second point B ; li B executes any other motion with reference
to a third point C ; and so on — the first is said to execute, with
reference to the last, a movement which is the resultant of these
several movements.
The relative position, velocity, and acceleration are in such a case
the geometrical resultants of the various components combined
according to preceding rules.
68. The following practical methods of effecting such a com-
bination' in the simple case of the movements of two points are
useful in scientific illustrations and in certain mechanical arrange-
ments. Let two moving points be joined by a uniform elastic string;
the middle point of this string will evidently execute a movement
which is half the resultant of the motions of the two points. But for
drawing, or engraving, or for other mechanical applications, the
following method is preferable : —
CF and ED are rods of equal length moving freely round a pivot
at /*, which passes through the middle point
of each— C^, AB, EB, and BE are rods of
half the length of the two former, and so
pivoted to them as to form a pair of equal
rhombi CD, EE, whose angles can be altered
at will. Whatever motions, whether in a plane,
or in space of three dimensions, be given to
A and B, /'will evidently be subjected to half
their resultant.
20 PRELIMINARY.
69. Amongst the most important classes of motions which we
have to consider in Natural Philosophy, there is one, namely, Har-
monic Motion^ which is of such immense use, not only in ordinary
kinetics, but in the theories of sound, light, heat, etc., that we make
no apology for entering here into some little detail regarding it.
70. Def. When a point Q moves uniformly in a circle, the per-
pendicular QP drawn from its position at any
instant to a fixed diameter AA' of the circle,
intersects the diameter in a point P^ whose
position changes by a simple harmo7iic motion.
Thus, if a planet or satellite, or one of the
constituents of a double star, be supposed to
move uniformly in a circular orbit about its
primary, and be viewed from a very distant
position in the plane of its. orbit, it will appear
to move backwards and forwards in a straight
line with a simple harmonic motion. This is nearly the case with
such bodies as the satellites of Jupiter when seen from the earth.
Physically, the interest of such motions consists in the fact of their
being approximately those of the simplest vibrations of sounding
bodies such as a tuning-fork or pianoforte-wire ; whence their name ;
and of the various media in which waves of sound, light, heat, etc.,
are propagated.
71. The Amplitude of a simple harmonic motion is the range on
one side or the other of the middle point of the course, i. e. OA or
OA' in the figure.
An arc of the circle referred to, or any convenient angular reck-
oning of it, measured from any fixed point to the uniformly moving
point Qy is the Argu??te?it of the harmonic motion.
[The distance of a point, performing a simple harmonic motion,
from the middle of its course or range, is a sifjiple harmonic functio7i
of the time; that is to say
a cos {lit - e),
where a, «, e are constants, and / represents time. The argument of
this function is what we have defined as the argument of the motion.
In the formula above, the argument is nt — e.'\
The Epoch in a simple harmonic motion is the interval of time
which elapses from the era of reckoning till the moving point first
comes to its greatest elongation in the direction reckoned as positive,
from its mean position or the middle of its range. [In the formula
above, put in the form
acosn
H}
e .
- is the epoch.] Epoch in angular measure is the angle described
n
on the circle of reference in the period of time defined as the epoch.
[In the formula, e is the epoch in angular measure.]
The Period of a simple harmonic motion is the time which elapses
KINEMATICS. 21
from any Instant until the moving point again moves in the same
direction through the same position, and is evidently the time of
revolution in the auxiliary circle. [In the formula the period is
27r -,
n '-'
The Phase of a simple harmonic motion at any instant is an
expression used to designate the part of its whole period which it
has reached. It is borrowed from the popular expression * phases of
the moon.' Thus for Simple Harmonic Motion we nlay call the
first or zero-phase that of passing through the middle position in the
positive direction. Then follow the successive phases quarter-period,
half-period, three-quarters-period, and complete period or return to
zero-phase. Sometimes it is convenient to reckon phase by a number
or numerical expression, which may be either a reckoning of angle or
a reckoning of time, or a fraction or multiple of the period. Thus
the positive maximum phase may sometimes be called the 90*^ phase
or the phase -, or the three-hour phase, if the period be 1 2 hours, or
the quarter-period phase. Or, again, the phase of half way down
from positive maximum may be described as the 120° phase or the
— phase, or the \ period phase. This particular way of specifying
phase is simply a statement of the argument as defined above and
measured from the point corresponding to positive motion through
the middle position.
72. Those common kinds of mechanism, for producing rectilineal
from circular motion, or vice versa, in which a, crank moving in
a circle works in a straight slot belonging to a body which can
only move in a straight line, fulfil strictly the definition of a simple
harmonic motion in the part of which the motion is rectilineal, if the
motion of the rotating part is uniform.
The motion of the treadle in a spinning-wheel approximates to
the same condition when the wheel moves uniformly ; the approxi-
mation being the closer, the smaller is the angular motion of the
treadle and of the connecting string. It is also approximated to
more or less closely in the motion of the piston of a steam-engine
connected, by any of the several methods in
use, with the crank, provided always the ro-
tatory motion of the crank be uniform.
73. The velocity of a point executing a
simple harmonic motion is a simple harmonic
function of the time, a quarter of a period
earlier in phase than the displacement, and
having its maximum value equal to the ve-
locity in the circular motion by which the given
function is defined.
For, in the fig., if F be the velocity in the circle, it may be
represented by OQ in a direction perpendicular to its own, and
22
PRELIMINARY.
therefore by OF and FQ in directions perpendicular to those lines.
That is, the velocity of F in the simple harmonic motion is
FQ V
YY7\ V or -?Yr) ^Q > which, when F passes through O, becomes V.
74. The acceleration of a point executing a simple hannonic
motion is at any time simply proportional to the displacement from
the middle point, but in opposite direction, or always towards the
middle point. Its maximum value is that with which a velocity
equal to that of the circular motion would be acquired in the time
in which an arc equal to the radius is described.
For m the fig., the acceleration of (2 (by § 36) is -^ along QO.
Supposing, for a moment, QO to represent the magnitude of this
acceleration, we may resolve it into QF, FO. The acceleration of
F is therefore represented on the same scale by FO. Its magnitude
V^ FO F-"
is therefore ^^•^^= ^j^^ FO, which is proportional to FO, and
has at A its maximum value, yr-^, an acceleration under which the
velocity V would be acquired in the time ~r- as stated. Thus we
have in simple harmonic motion
Acceleration _ F^ _47r^
Displacement ~ Q^^ T^
where T is the time of describing the circle, or the period of the
harmonic motion.
75. Any two simple harmonic motions in one line, and of one
period, give, when compounded, a single simple harmonic motion ;
of the same period; of amplitude equal
to the diagonal of a parallelogram de-
scribed on lengths equal to their am-
plitudes measured on lines meeting at
an angle equal to their difference of
epochs; and of epoch differing from
their epochs by angles equal to those
which this diagonal makes with the
two sides of the parallelogram. Let F
and jP' be two points executing simple
harmonic motions of one period, and in
one line B'BCAA'. Let Q and Q be the
uniformly moving points in the relative
circles. On CQ and CQ describe a
parallelogram SQCQ'-, and through S draw SR perpendicular to
FA' produced. We have F'R= CP (being projections of the equal
and parallel lines QS, CQ, on CR). Hence CR^CF-hCF';
and therefore the point R executes the resultant of the motions F
and F'. But CS, the diagonal of the parallelogram, is constant
KINEMATICS. 23
(since the angular velocities of CQ and CQ are equal, and therefore
the angle QCQ' is constant), and revolves with the same angular
velocity as CQ or CQ' ; and therefore the resultant motion is simple
harmonic, of amplitude CS, and of epoch exceeding that of the
motion of F, and falling short of that of the motion of F'y by the
angles (2^15" and SCQ' respectively.
This geometrical construction has been usefully applied by the
tidal committee of the British Association for a mechanical tide-
indicator (compare § 77 below). An arm CQ turning round C
carries an arm QS turning round Q. Toothed wheels, one of them
fixed with its axis through C, and the others pivoted on a framework
carried by CQ, are so arranged that QS turns very approximately at
the rate of once round in 1 2 mean lunar hours, if CQ be turned uni-
formly at the rate of once round in 1 2 mean solar hours. Days and
half-days are marked by a counter geared to CQ. The distance of
S from a fixed line through C shows the deviation from mean sea-
level due to the sum of mean solar and mean lunar tides for the time
of day and year marked by CQ and the counter.
76. The construction described in the preceding section exhibits
the resultant of two simple harmonic motions, whether of the same
period or not. [If they are very nearly, but not exactly, of the same
period, the diagonal of the parallelogram will not be constant, but
will diminish from a maximum value, the sum of the component
amplitudes, which it has at the instant when the phases of the
component motions agree ; to a minimum, the difi"erence of those
amplitudes, which is its value when the phases differ by half a period.
Its direction, which always must be nearer to the greater than to the
less of the two radii constituting the sides of the parallelogram, will
oscillate on each side of the greater radius to a maximum deviation
amounting on either side to the angle whose sine is the less radius
divided by the greater, and reached when the less radius deviates by
this together with a quarter circumference, from the greater. The
full period of this oscillation is the time in which either radius gains
a full turn on the other. The resultant motion is therefore not
simple harmonic, but is, as it were, simple harmonic with periodi-
cally increasing and diminishing amplitude, and with periodical
acceleration and retardation of phase. This view is most appropriate
for the case in which the periods of the two component motions
are nearly equal, but the amplitude of one of them much greater than
that of the other.
To find the amount of the maximum advance and maximum back-
wardness of phase, and when they are experienced, let CA be equal
to the greater half-amplitude. From A as
centre, with AB the less half-amplitude as
radius, describe a circle. CB touching this
circle represents the most deviated resultant.
Hence CBA is a right angle ; and
smBCA=^.
24 PRELIMINARY.
The angle BCA thus found is the amount by which the phase of
the resultant motion is advanced or retarded relatively to that of the
larger component; and the supplement of BCA is the difference of
phase of the two components at the time of maximum advance or
backwardness of the resultant.]
77. A most interesting application of this case of the composition
of harmonic motions is to the lunar and solar tides ; which, except
in tidal rivers, or long channels or bays, follow each very nearly the
simple harmonic law, and produce, as the actual result, a variation
of level equal to the sum of variations that would be produced by
the two causes separately.
The amount of the lunar tide in the equilibrium theory is about
2*1 times that of the solar. Hence the spring tides of this theory
are 3-1, and the neap tides only i-i, each reckoned in terms of the
solar tide; and at spring and neap tides the hour of high water is
that of the lunar tide alone. The greatest deviation of the actual
tide from the phases (high, low, or mean water) of the lunar tide
alone, is about -95 of a lunar hour, that is, -98 of a solar hour (being
the same part of 12 lunar hours that 28° 26', or the angle whose
sine is -7-, is of 360°). This maximum deviation will be in advance
or in arrear according as the crown of the solar tide precedes or
follows the crown of the lunar tide; and it will be exactly reached
when the interval of phase between the two component tides is 3*95
lunar hours. That is to say, there will be maximum advance of the
time of high water approximately 4I days after, and maximum retar-
dation the same number of days before, spring tides.
78. We may consider next the case of equal amplitudes in the
two given motions. If their periods are equal, their resultant is a
simple harmonic motion, whose phase is at every instant the mean
of their phases, and whose amplitude is equal to twice the amph-
tude of either multiplied by the cosine of half the difference of
their phase. The resultant is of course nothing when their phases
differ by half the period, and is a motion of double amplitude and
of phase the same as theirs when they are of the same phase.
When their periods are very nearly, but not quite, equal (their
amphtudes being still supposed equal), the motion passes very slowly
from the former (zero, or no motion at all) to the latter, and back,
in a time equal to that in which the faster has gone once oftener
through its period than the slower has.
In practice we meet with many excellent examples of this case,
which will, however, be more conveniently treated of when we come
to apply kinetic principles to various subjects in practical mechanics,
acoustics, and physical optics ; such as the marching of troops over a
suspension bridge, the sympathy of pendulums or tuning-forks, etc.
79. If any number of pulleys be so placed that a cord passing
from a fixed point half round each of them has its free parts all
in parallel lines, and if their centres be moved with simple harmonic
KINEMATICS. 25
motions of any ranges and any periods in lines parallel to those
lines, the unattached end of the cord moves with a complex har-
monic motion equal to twice the sum of the given simple har-
monic motions. This is the principle of Sir W. Thomson's tide-
predicting machine, constructed by the British Association, and order-
ed to be placed in South Kensington Museum, availably for general
use in calculating beforehand for any port or other place on the sea
for which the simple harmonic constituents of the tide have been de-
termined by the 'harmonic analysis' applied to previous observa-
tions \ We may exhibit, graphically, the various preceding cases of
single or compound simple harmonic motions in one line by curves
in which the abscissae represent intervals of time, and the ordinates
the corresponding distances of the moving point from its mean
position. In the case of a single simple harmonic motion, the
corresponding curve would be that described by the point P in
§ 70, if, Avhile Q maintained its uniform circular motion, the circle
were to move with uniform velocity in any direction perpendicular
to OA. This construction gives the harmonic curve, or curve of
sines, in which the ordinates are proportional to the sines of the
abscissae, the straight line in which O moves being the axis of
abscissae. It is the simplest possible form assumed by a vibrating
string ; and when it is assumed that at each instant the motion
of every particle of the string is simple harmonic. When the
harmonic motion is complex, but in one line, as is the case
for any point in a violin-, harp-, or pianoforte-string (differing,
as these do, from one another in their motions on account
of the different modes of excitation used), a similar construction
may be made. Investigation regarding complex harmonic functions
has led to results of the highest importance, having their most
general expression in Fourier's Theorem^ to be presently enunciated.
We give below a graphic representation of the composition of two
simple harmonic motions in one line, of equal amplitudes and of
periods which are as i : 2 and as 2 13, the epochs being each
a quarter circumference. The horizontal line is the axis of ab-
scissae of the curves ; the vertical line to the left of each being the
axis of ordinates. In the first case the slower motion goes through
1:2 2:3
(Octave) (Fifth)
^ .
1 See British Association Tidal Committee's Reports, 1868, 1872, 1875 • ^"^
Lecture on Tides, by Sir \V. Thomson (Collins, Glasgow, 1876).
26
PRELIMINARY.
one complete period, in the second it goes through two periods.
These and similar cases when the periodic times are not commen-
surable, will be again treated of under Acoustics.
80. We have next to consider the composition of simple har-
monic motions in different directions. In the first place, we see
that any number of simple harmonic motions of one period, and
of the same phase, superimposed, produce a single simple harmonic
motion of the same phase. For, the displacement at any instant
being, according to the principle of the composition of motions, the
geometrical resultant of the displacements due to the component
motions separately, these component displacements in the case sup-
posed, all vary in simple proportion to one another, and are in
constant directions. Hence the resultant displacement will vary
in simple proportion to each of them, and will be in a constant
direction.
But if, while their periods are the same, the phases of the several
component motions do not agree, the resultant motion will generally
be elliptic, with equal areas described in equal times by the radius-
vector from the centre ; although in particular cases it may be uni-
form circular, or, on the other hand, rectilineal and simple harmonic.
81. To prove this, we may first consider the case, in which we
have two equal simple harmonic motions given, and these in per-
pendicular lines, and differing in phase by a quarter period. Their
resultant is a uniform circular motion. For, let BA^ B'A' be their
ranges ; and from 6>, their common middle point as centre, describe
a circle through AA' BB'. The given motion
of P in BA will be (§67) defined by the
motion of a point Q, round the circumference
of this circle ; and the same point, if moving
in the direction indicated by the arrow, will
give a simple harmonic motion of P', in
BA\ a quarter of a period behind that of
the motion oi P in BA. But, since A'OA,
QPO, and QP'O are right angles, the figure
QPOP is a parallelogram, and therefore Q is in the position of the
displacement compounded of OP and OP'. Hence two equal simple
harmonic motions in perpendicular lines, of phases differing by a
quarter period, are equivalent to a uniform circular motion of radius
equal to the maximum displacement of either singly, and in the direc-
tion from the positive end of the range of the component in advance
of the other towards the positive end of the range of this latter.
82. Now, orthogonal projections of simple harmonic motions are
clearly simple harmonic with unchanged phase. Hence, if we pro-
ject the case of § 81 on any plane, we get motion in an ellipse, of
which the projections of the two component ranges are conjugate
diameters, and in which the radius-vector from the centre describes
equal areas (being the projections of the areas described by the
radius of the circle) in equal times. But the plane and position of
- KINEMATICS. 27
the circle of which this projection is taken may clearly be found so
as to fulfil the condition of having the projections of the ranges
coincident with any two given mutually bisecting lines. Hence any
two given simple harmonic motions, equal or unequal in range, and
oblique or at right angles to one another in direction, provided only
they differ by a quarter period in phase, produce elliptic motion,
having their ranges for conjugate axes, and describing, by the
radius-vector from the centre, equal areas in equal times.
83. Returning to the composition of any number of equal simple
harmonic motions in lines in all directions and of all phases : each
component simple harmonic motion may be determinately resolved
into two in the same line, differing in phase by a quarter period,
and one of them having any given epoch. We may therefore reduce
the given motions to two sets, differing in phase by a quarter period,
those of one set agreeing in phase with any one of the given, or
with any other simple harmonic motion we please to choose (i.e.
having their epoch anything we please).
All of each set may (§ 75) be compounded into one simple har-
monic motion of the same phase, of determinate amplitude, in a de-
terminate line ; and thus the whole system is reduced to two simple
fully-determined harmonic motions differing from one another in
phase by a quarter period.
Now the resultant of two simple harmonic motions, one a quarter
of a period in advance of the other, in different lines, has been
proved (§ 82) to be motion in an ellipse of which the ranges of the
component motions are conjugate axes, and in which equal areas
are described by the radius-vector from the centre in equal times.
Hence the proposition of § 80.
84. We must next take the case of the composition of simple
harmonic motions of different periods and in different lines. In
general, whether these lines be in one plane or not, the line of
motion returns into itself if the periods are commensurable ; and if
not, not. This is evident without proof
Also we see generally that the composition of any number of
simple harmonic motions in any directions and of any periods, may
be effected by adding their components in each of any three rect-
angular directions. The final resultant motion is thus fully expressed
by formulae giving the rectangular co-ordinates as 'complex harmonic
functions ' of the time.
85. By far the most interesting case, and by far the simplest,
is that of two simple harmonic motions of any periods, whose
directions must of course be in one plane.
Mechanical methods of obtaining such combinations will be after-
wards described, as well as cases of their occurrence in Optics and
Acoustics.
We may suppose, for simplicity, the two component motions to take
place in perpendicular directions. Also, it is easy to see that we can
only have a reentering curve when their periods are commensurable.
28
PRELIMINARY,
The following figures represent the paths produced by the com-
bination of simple harmonic motions of equal amplitude in two rect-
angular directions, the periods of the components being as i : 2,
and the epochs differing successively by o,
a circumference.
I, 2, etc., sixteenths of
In the case of epochs equal, or differing by a multiple of tt, the
curve is a portion of a parabola, and is gone over twice in opposite
directions by the moving point in each complete period.
If the periods be not exactly as i : 2 the form of the path pro-
duced by the combination changes gradually from one to another
of the series above figured ; and goes through all its changes in the
time in which one of the components gains a complete vibration on
the other.
86. Another very important case is that of two pairs of simple
harmonic motions in one plane, such that the resultant of each pair
is uniform circular motion.
If their periods are equal, we have a case belonging to those
already treated (§ 80), and conclude that the resultant is, in general,
motion in an ellipse, equal areas being described in equal times
about the centre. As particular cases we may have simple har-
monic, or uniform circular, motion.
If the circular motions are in the sajtie direction, the resultant is
evidently circular motion in the same direction. This is the case
of the motion of ^ in § 75, and requires no further comment, as
its amplitude, epoch, etc., are seen at once from the figure.
KINEMATICS. 29
87. If the radii of the component motions are equal, and the
periods very nearly equal, but the motions in opposite directions,
we have cases of great importance in modern physics, one of which
is figured below (in general, a non-reentrant curve).
This is intimately connected with the explanation of two sets of
important phenomena, — the rotation of the plane of polarization of
light, by quartz and certain fluids on the one hand, and by trans-
parent bodies under magnetic forces on the other. It is a case of
the hypotrochoid, and its corresponding mode of description will be
described in § 104. It may be exhibited experimentally as the path
of a pendulum, hung so as to be free to move in any vertical plane
tlirough its point of suspension, and containing in its bob a fly-wheel
in rapid rotation.
88. [Before leaving for a time the subject of the composition of
harmonic motions, we must enunciate Fourier's Theorem, which is
not only one of the most beautiful results of modern analysis, but
may be said to furnish an indispensable instrument in the treatment
of nearly every recondite question in modern physics. To mention
only sonorous vibrations, the propagation of electric signals along
a telegraph wire, and the conduction of heat by the earth's crust,
as subjects in their generality intractable without it, is to give but
a feeble idea of its importance. Unfortunately it is impossible to
give a satisfactory proof of it without introducing some rather trouble-
some analysis, which is foreign to the purpose of so elementary a
treatise as the present.
The following seems to be the most intelligible form in which it
can be presented to the general reader r —
Theorem. — A complex harmonic function^ with a constant term
added, is the proper expression, in mathematical langiiage, for any
arbitrary periodic function ; and consequently can express any function
whatever between definite values of the variable.
30 PRELIMINARY.
89. Any arbitrary periodic function whatever being given, the
amplitudes and epochs of the terms of a complex harmonic function,
which shall be equal to it for every value of the independent variable,
may be investigated by the ' method of indeterminate coefficients.'
Such an investigation is sufficient as a solution of the problem, — to
find a complex harmonic function expressing a given arbitrary
periodic function, — when once we are assured that the problem is
possible ; and when we have this assurance, it proves that the reso-
lution is determinate ; that is to say, that no other complex harmonic
function than the one we have found can satisfy the conditions.]
90. We now pass to the consideration of the displacement of a
rigid body or group of points whose relative positions are unalterable.
The simplest case we can consider is that of the motion of a plane
figure in its own plane, and this, as far as kinematics is concerned,
is entirely summed up in the result of the next section.
91. If a plane figure be displaced in any way in its own plane,
there is always (with an exception treated in § 93) one point of it
common to any two positions ; that is, it may be moved from any
one position to any other by rotation in its own plane about one
point held fixed.
To prove this, let A^ B be any two points of the plane figure in a
first position, A\ B' the position of the same two after a displacement.
The Hnes AA\ BB' will not be parallel, except in one case to be
presently considered. Hence the Hne equidistant from A and A'
will meet that equidistant from B and B' in some point O. Join
OA, OB, OA', OB'. Then, evidently, because OA' = OA, OB' = OB,
and A'B' = AB, the triangles OA'B' and OAB are equal and similar.
Hence O is similarly situated with regard to A'B' and AB, and
is therefore one and the same point of the plane figure in its
two positions. If, for the sake of illustration,
^B we actually trace the angle OAB upon the
plane, it becomes OA'B' in the second posi-
tion of the figure.
92. If from the equal angles A' OB', A OB
of these similar triangles we take the com-
mon part A' OB, we have the remaining
angles AOA', BOB' equal, and each of them
is clearly equal to the angle through which
the figure must have turned round the point O
to bring it from the first to the second position.
The preceding simple construction therefore enables us not only
to demonstrate the general proposition (§91), but also to determine
from the two positions of one line AB, A'B' of the figure the
common centre and the amount of the angle of rotation.
93. The lines equidistant from A and A', and from B and B',
are parallel if ^^ is parallel to A'B' ) and therefore the construction
KINEMATICS.
31
A
fails, the point O being infinitely-
distant, and the theorem becomes
nugatory. In this case the motion is
in fact a simple translation of the
figure in its own plane without rota-
tion— since as AB is parallel and equal
to A'B\ we have A A' parallel and equal to BB' \ and instead of
there being one point of the figure common to both positions, the
lines joining the successive positions of every point in the figure are
equal and parallel.
94. It is not necessary to suppose the figure to be a mere flat
disc or plane — for the preceding statements apply to any one of a
set of parallel planes in a rigid body, moving in any way subject to
the condition that the points of any one plane in it remain always in
a fixed plane in space.
95. There is yet a case in which the construction in §
nugatory — that is when A A' is parallel
to BB', but AB intersects A B'. In
this case, however, it is easy to see at
once that this point of intersection is the
point O required, although the former
method would not have enabled us to
find it.
96. Very many interesting applications of this principle may be
made, of which, however, few belong strictly to our subject, and we
shall therefore give only an example or two. Thus we know that
if a line of given length AB move with its extremities always in two
fixed lines OA^ OB, any point in it as P describes an ellipse. (This
is proved in § 101 below.) It is required to find the direction of
motion of P at any instant, i. e. to draw a tangent to the ellipse.
BA will pass to its next position by rotating about the point Q ; found
by the method of § 91 by drawing per-
pendiculars to OA and OB at A and
B. Hence P for the instant revolves
about Qj and thus its direction of
motion, or the tangent to the ellipse,
is perpendicular to QP. Also AB in
its motion always touches a curve
(called in geometry its envelop) ; and
the same principle enables us to find
the point of the envelop which lies in
AB, for the motion of that point must 0
evidently be ultimately (that is for a very small displacement) along
AB, and the only point which so moves is the intersection of AB,
with the perpendicular to it from Q. Thus our construction would
enable us to trace the envelop by points.
3 2 PRELIMINAR K
97. Again, suppose ABDC to be a jointed frame, AB having
a reciprocating motion about A^ and by a link ^Z> turning CD in
the same plane about C. Deter-
■4 y ^ mine the relation between the
angular velocities of AB and
CD in any position. Evidently
the instantaneous direction of
motion of B is transverse to
AB^ and oi D transverse to CD —
hence if AB, CD produced meet
in O, the motion of BD is for an instant as if it turned about O.
From this it may easily be seen that if the angular velocity of AB
be <o, that of CD is -^^-^ -j^^=- w. A similar process is of course
UJd CD
applicable to any combination of machinery, and we shall find it very
convenient when we come to apply the principle of work in various
problems of Mechanics.
Thus in any Lever, turning in the plane of its arms — the rate of
motion of any point is proportional to its distance from the fulcrum,
and its direction of motion at any instant perpendicular to the line
joining it with the fulcrum. This is of course true of the particular
form of lever called the Wheel and Axle.
98. Since, in general, any movement of a plane figure in its plane
may be considered as a rotation about one point, it is evident that
two such rotations may, in general, be compounded into one ; and
therefore, of course, the same may be done with any number of
rotations. Thus let A and B be the points of the figure about which
in succession the rotations are to take place. By rotation about
A, B is brought say to B, and by a rotation about B' , A is brought
to A'. The construction of § 91 gives us at once the point O and
the amount of rotation about it which singly gives the same effect
as those about A and B in succession. But there is one case of
exception, viz. when the rotations about A and B are of equal
amount and in opposite directions. In
this case A' B' is evidently parallel and
equal to AB, and therefore the com-
pound result is a tra?islation only. Thus
we see that if a body revolve in succes-
sion through equal angles, but in oppo-
site directions, about two parallel axes, it finally takes a position
to which it could have been brought by a simple translation per-
pendicular to the lines of the body in its initial or final position,
which were successively made axes of rotation ; and inclined to their
plane at an angle equal to half the supplement of the common angle
of rotation.
99. Hence to compound into an equivalent rotation a rotation
and a translation, the latter being effected parallel to the plane of
the former, we may decompose the translation into two rotations
KINEMATICS. 33
of equal amounts and opposite directions, compound one of them
with the given rotation by § 98, and then compound the other with
the resultant rotation by the same process. Or we may adopt the
following far simpler method : — Let
OA be the translation common to
all points in the plane, and let
BOC be the angle of rotation k^, ' 0
about O, BO being drawn so that JB
OA bisects the exterior angle COB'. Evidently there is a point
B' m. BO produced, such that B'C\ the space through which the
rotation carries it, is equal and opposite to OA, This point retains
its former position after the performance of the compound operation ;
so that a rotation and a translation in one plane can be compounded
into an equal rotation about a different axis.
100. Any motion whatever of a plane figure in its own plane
might be produced by the rolling of a curve fixed to the figure
upon a curve fixed in the plane.
For we may consider the whole motion as made up of successive
elementary displacements, each of which corresponds, as we have
seen, to an elementary rotation about some
point in the plane. Let 6>,, O^, O^, etc.,
be the successive points of the figure about
which the rotations take place, 0^, o^, 0.^,
etc., the positions of these points on the
plajie when each is the instantaneous centre
of rotation. Then the figure rotates about
(9j (or ^j, which coincides with it) till O^
coincides with 0^^ then about the latter till
O^ coincides with ^3, and so on. Hence, if
we join (9^, O^, O3, etc., in the plane of the figure, and ^,, 0^, 0^, etc.,
in the fixed plane, the motion will be the same as if the polygon
O^O^O^, etc., rolled upon the fixed polygon o^o^o^, etc. By
supposing the successive displacements small enough, the sides
of these polygons gradually diminish, and the polygons finally
become continuous curves. Hence the theorem.
From this it immediately follows, that any displacement of a rigid
solid, which is in directions wholly perpendicular to a fixed hne,
may be produced by the rolling of a cylinder fixed in the solid on
another cylinder fixed in space, the axes of the cylinders being
parallel to the fixed line.
101. As an interesting example of this theorem, let us recur to the
case of § 96 : — A circle may evidently be circumscribed about OBQA;
and it must be of invariable magnitude, since in it a chord of given
length AB subtends a given angle O at the circumference. Also OQ is
a diameter of this circle, and is therefore constant. Hence, as Q is
momentarily at rest, the motion of the circle circumscribing OBQA
is one of internal rolling on a circle of double its diameter. Hence
if a circle roll internally on another of twice its diameter any point in
T. X
34
PRELIMINARY.
its circumference describes a dianieter of the fixed circle, any other
point in its plane an ellipse. This is precisely the same proposition
as that of § 86, although the ways of arriving at it are very different.
102. We may easily employ this result, to give the proof, promised
in § 96, that the point P oi AB describes an ellipse. Thus let
OAy OB be the fixed lines, in which the extremities of AB move.
Draw the circle A OBD, circumscribing A OB, and let CD be the
diameter of this circle which passes through P. While the two
points A and B of this circle move along OA and OB, the points
C and D must, because of the invariability of the angles BOD,
AOC, move along straight lines OC,
OD, and these are evidently at right
angles. Hence the path of P may
be considered as that of a point in
a line whose ends move on two
mutually perpendicular lines. Let E
be the centre of the circle ; join OE,
and produce it to meet, in F, the
line FPG drawn through P parallel
to DO. Then evidently EF^EP,
hence F describes a circle about O.
Also FP : EG :: 2FE : FO, or PG
is a constant submultiple oi EG; and
therefore the locus of P is an ellipse
whose major axis is a diameter of the
circular path oi F. Its semi-axes are DP dXong OC, and i^C along OD.
103. When a circle rolls upon a straight line, a point in its
circumference describes a Cycloid, an internal point describes a
Prolate Cycloid, an external point a Curtate Cycloid. The two
latter varieties are sometimes called Trochoids.
The general form of these curves will be seen in the succeeding
figures; and in what follows we shall confine our remarks to the
cycloid itself, as it is of greater consequence than the others. The
next section contains a simple investigation of those properties of
the cycloid which are most useful in our subject.
i
104. Let AB be a diameter of the generating (or rolling) circle,
BC the line on which it rolls. The points A and B describe similar
and equal cycloids, of which AQC and ^^S* are portions, li FQR
be any subsequent position of the generating circle, Q and S the
new positions of A and B, QFS is of course a right angle. If,
therefore, QR be drawn parallel to
FS, FR is a diameter of the rolling
circle, and R lies in a straight line
AH drawn parallel to BC. Thus
AR = BF. Produce QR to 7]
making FT= QR^FS. Evidently
the curve AT, which is the locus
of T, is similar and equal to BS,
and is therefore a cycloid similar and ^
equal to AC. But QR is perpen-
dicular to FQ, and is therefore the
instantaneous direction of motion of
Q, or is the tangent to the cycloid
AQC. Similarly, FS is perpendicular to the cycloid ^^S* at S, and
therefore TQ is perpendicular to ^7" at T. Hence ($22) AQC is
the evolute oi AT, and 2.rc AQ= QT=2QR,
105. When a circle rolls upon an-
other circle, the curve described by a
point in its circumference is called an
Epicycloid, or a Hypocycloid, as the
rolling circle is without or within the
fixed circle; and when the tracing-point
is not in the circumference, we have
Epitrochoids and Hypotrochoids. Of
the latter classes we have already met
with examples (§§ 87, loi), and others
will be presently mentioned. Of the
former we have, in the first of the
appended figures, the case of a circle
rolling externally on another of equal
size. The curve in this case is called
the Cardioid.
In the second figure a circle rolls ex-
36
PRELIMINARY.
ternally on another of twice its radius. The epicycloid so described
is of importance in optics, and will, with others, be referred to when
we consider the subject of Caustics by reflexion.
In the third figure we have a hypo-
cycloid traced by the rolling of one
circle internally on another of four
times its radius.
The curve of § 87 is a hypotrochoid
described by a point in the plane of a
circle which rolls internally on another
of rather more than twice its diameter,
the tracing-point passing through the
centre of the fixed circle. Had the
diameters of the circles been exactly as
I : 2, § loi shows us that this curve
would have been reduced to a single straight line.
106. If a rigid body move in any way whatever, subject only to
the condition that one of its points remains fixed, there is always
(without exception) one line of it through this point common to the
body in any two positions.
Consider a spherical surface within the body, with its centre at the
fixed point C. All points of this sphere attached to the body will
move on a sphere fixed in space. Hence the construction of § 91
may be made, only with great circles instead of straight lines ;
and the same reasoning will apply to prove that the point O thus
obtained is common to the body in its two positions. Hence every
point of the body in the line (9C, joining O with the fixed point,
must be common to it in the two positions. Hence the body may
pass from any one position to any other by a definite amount of
rotation about a definite axis. And hence, ako, successive or simul-
taneous rotations about any number of axes through the fixed point
may be compounded into one such rotation.
107. Let OA^ OB be two axes about which a body revolves with
angular velocities o>, w^ respectively.
With radius unity describe the arc AB^ and in it take any point /.
Draw /a, //? perpendicular to OA, OB respectively. Let the rota-
tions about the two axes be such that that about
OB tends to raise I above the plane of the
paper, and that about OA to depress it. In an
infinitely short interval of time r, the amounts of
these displacements will be Wj/^ . t and — oo/a . t.
The point /, and therefore every point in the
line 01, will be at rest during the interval t if
the sum of these displacements is zero — i.e. if
o)j . //3 = (0 . 7a.
Hence the line 01 is instantaneously at rest, or the two rotations
about OA and OB may be compounded into one about 01. Draw
KINEMA TICS. 3 7
Ip, Iq, parallel to OB, OA respectively. Then, expressing in two
ways the area of the parallelogram IpOq, we have
Oq,ip = Op.Ia.
Hence Oq -. Op :: m^ -. w.
In words, if on the axes OA, OB, we measure off from O lines
Op, Oq, proportional respectively to the angular velocities about
these axes — the diagonal of the parallelogram of which these are
contiguous sides is the resultant axis.
Again, if Bb be drawn perpendicular to OA, and if O be the
angular velocity about 01, the whole displacement oi B may evidently
be represented either by
oi.Bbora. 7/3.
Hence n : oi :: Bb : I^
:: 0/ : Op.
And thus on the scale on which Op, Oq represent the component
angular velocities, the diagonal 01 represents their resultant.
108. Hence rotations are to be compounded according to the
same law as velocities, and therefore the single angular velocity,
equivalent to three co-existent angular velocities about three mutually
perpendicular axes, is determined in magnitude, and the direction
of its axis is found, as follow* : — The square of the resultant angular
velocity is the sum of the squares of its components, and the ratios
of the three components to the resultant are the direction-cosines
of the axis.
Hence also, an angular velocity about any line may be resolved
into three about any set of rectangular lines, the resolution in each
case being (like that of simple velocities) effected by multiplying by
the cosine of the angle between the directions.
Hence, just as in § 38 a uniform acceleration, acting perpendi-
cularly to the direction of motion of a point, produces a change in
the direction of motion, but does not influence the velocity; so, if
a body be rotating about an axis, and be subjected to an action
tending to produce rotation about a perpendicular axis, the result
will be a change of direction of the axis about which the body
revolves, but no change in the angular velocity. On this kinematical
principle is founded the dynamical explanation of the precession of
the equinoxes, and some of the seemingly marvellous performances
of gyroscopes and gyrostats.
109. If a pyramid or cone of any form roll on a similar pyramid
(the image in a plane mirror of the first position of the first) all
round, it clearly comes back to its primitive position. This (as all
rolling of cones) is exhibited best by taking the intersection of each
with a spherical surface. Thus we see that if a spherical polygon
turns about its angular points in succession, always keeping on the
spherical surface, and if the angle through which it turns about each
point is twice the supplement of the angle of the polygon, or, which
38 PRELIMINARY.
will come to the same thing, if it be in the other direction, but
equal to twice the angle itself of the polygon, it will be brought to
its original position,
110. The method of § loo also applies to the case of § io6; and
it is thus easy to show that the most general motion of a spherical
figure on a fixed spherical surface is obtained by the rolling of a
curve fixed in the figure on a curve fixed on the sphere. Hence as
at each instant the line joining C and O contains a set of points of
the body which are momentarily at rest, the most general motion of
a rigid body of which one point is fixed consists in the rolling of a
cone fixed in the body upon a cone fixed in space — the vertices of
both being at the fixed point.
111. To complete our kinematical investigation of the motion of
a body of which one point is fixed, we require a solution of the fol-
lowing problem: — From the given angular velocities of the body
at each instant about three rectangular axes attached to it to de-
termine the position of the body in space after a given time. But
the general solution of this problem demands higher analysis than
can be admitted into the present treatise.
112. We shall next consider the most general possible motion of
a rigid body of which no point is fixed — and first we must prove
the following theorem. There is on^ set of parallel planes in a
rigid body which are parallel to each other in any two positions of
the body. The parallel lines of the body perpendicular to these
planes are of course parallel to each other in the two positions.
Let C and C be any point of the body in its first and second
positions. Move the body without rotation from its second position
to a third in which the point at C in the second position shall
occupy its original position C. The preceding demonstration shows
that there is a line CO common to the body in its first and third
positions. Hence a line CO' of the body in its second position is
parallel to the same line CO in the first position. This of course
clearly applies to every line of the body parallel to CO, and the
planes perpendicular to these lines also remain parallel.
113. Let S denote a plane of the body, the two positions of which
are parallel. Move the body from its first position, without rotation,
in a direction perpendicular to *S, till S comes into the plane of its
second position. Then to get the body into its actual position, such
a motion as is treated in § 91 is farther required. But by § 91 this
may be effected by rotation about a certain axis perpendicular to the
plane *S, unless the motion required belongs to the exceptional case
of pure translation. Hence (this case excepted), the body may
be brought from the first position to the second by translation
through a determinate distance perpendicular to a given plane, and
rotation through a determinate angle about a determinate axis per-
pendicular to that plane. This is precisely the motion of a scrc^v
in its nut.
KINEMATICS. 39
114. To understand the nature of this motion we may com-
mence with the shding of one straight-edged board on another.
Thus let GDEF be a plane board whose edge, DE, sUdes on
the edge, AB, of another board, ABC, of which for convenience
we suppose the edge, AC, to be hori-
zontal. By § 30, if the upper board
move horizontally to the right, the
constraint will give it, in addition, a
vertically upward motion, and the rates
of these motions are in the constant
ratio of ^C to CB. Now, if both
planes be bent so as to form portions
of the surface of a vertical right cylinder, the motion of DF parallel
to ^C will become a rotation about the axis of the cylinder, and
the necessary accompaniment of vertical motion will remain un-
changed. As it is evident that all portions of AB will be equally
inclined to the axis of the cylinder, it is obvious that the thread
of the screw, which corresponds to the edge, £>E, of the upper
board, must be traced on the cylinder so as always to make a con-
stant angle with its generating lines (§ 128). A hollow mould
taken from the screw itself forms what is called the nut — the re-
presentative of the board, ABC — and it is obvious that the screw
cannot move without rotating about its axis, if the nut be fixed.
If a be the radius of the cylinder, w the angular velocity, a the
inclination of the screw thread to a generating line, u the linear
velocity of the axis of the screw, we see at once from the above con-
struction that
aia : u \\ AC ; CB :: sin a : cos a,
which gives the requisite relation between o> and u.
115. In the excepted case of § 113, the whole motion consists
of two translations, which can of course be compounded into a
single one : and thus, in this case, there is no rotation at all, or
every plane of it fulfils the specified condition for *S of § 113.
116. We may now briefly consider the case in which the guiding
cones (§ 110) are both circular, as it has important applications to
the motion of the earth, the evolutions of long or flattened projec-
tiles, the spinning of tops and gyroscopes, etc. The motion in this
case may be called Frecessioiial Rotation. The plane through the
instantaneous axis and the axis of the fixed cone passes through the
axis of the rolling cone. This plane turns round the axis of the
fixed cone with an angular velocity O, which must clearly bear a
constant ratio to the angular velocity w of the rigid body about its
instantaneous axis.
117. The motion of the plane containing these axes is called the
precession in any such case. What we have denoted by O is the
angular velocity of the precession, or, as it is sometimes called, the
rate of precession.
40 PRELIMINARY.
The angular motions w, O are to one another inversely as the
distances of a point in the axis of the rolling cone from the in-
stantaneous axis and from the axis of the fixed cone.
For, let OA be the axis of the fixed cone, OB that of the rolling
cone, and 01 the instantaneous axis. From any point P in OB
draw PN perpendicular to 01, and PQ perpendicular to OA.
Then we perceive that P moves always in the circle whose centre
is (2, radius PQ, and plane perpendicular
to OA. Hence the actual velocity of
the point /^ is 12 . QP. But, by the prin-
ciples explained above (§ no) the velocity
, of jP is the same as that of a point moving
in a circle whose centre is N, plane per-
pendicular to ON, and radius NP, which,
as this radius revolves with angular velo-
city CO, is 0) . NP. Hence
0 n.QP=i^.NP,
• or id '. Q, V. QP : NP.
118. Suppose a rigid body bounded by any curved surface to be
touched at any point by another such body. Any motion of one
on the other must be of one or more of the forms sliding, rollings
or spin7iing. The consideration of the first is so simple as to require
no comment.
Any motion in which the bodies have no relative velocity at the
point of contact, must be rolling or spinning, separately or combined.
Let one of the bodies rotate about successive instantaneous axes,
all lying in the common tangent plane at the point of instantaneous
contact, and each passing through this point — the other body being
fixed. This motion is what we call rolling, or simple rolling, of the
movable body on the fixed.
On the other hand, let the instantaneous axis of the moving body
be the common normal at the point of contact. This is pure spin-
ning, and does not change the point of contact.
Let it move, so that the instantaneous axis, still passing through
the point of contact, is neither in, nor perpendicular to, the tangent
plane. This motion is combined rolling and spinning.
119. As an example of pure rolling, we may take that of one
cylinder on another, the axes being parallel.
Let p be the radius of curvature of the rolling, o- of the fixed,
cylinder ; w the angular velocity of the former, V the linear velocity
of the point of contact. We have
C*3
For, in the figure, suppose P to be at any time the point of
contact, and Q and p the points which are to be in contact after
a very small interval t ; O, O the centres of curvature ; POp = 6,
PO'Q = cf>.
KINEMATICS.
41
Then ^(2 = i^ = space described by point of con-
tact. In symbols
p<^ = o-6»= Vr.
Also, before 0'Q.2.Xi^ OP can coincide in direc-
tion, the former must evidently turn through an angle
Therefore wt = ^ + <^ ;
and by eliminating B and ^, and dividing by t, we
get the above result.
It is to be understood here, that as the radii of
curvature have been considered positive when both
surfaces are convex, the negative sign must be intro-
duced for either radius when the corresponding sur-
face is concave.
Hence the angular velocity of the rolling curve is in this case
equal to the product of the linear velocity of the point of contact into
the sum or difference of the curvatures, according as the curves are
both convex, or one concave and the other convex.
120. We may now take up a few points connected with the curva-
ture of surfaces, which are useful in various parts of our subject.
The tangent plane at any point of a surface may or may not cut
it at that point. In the former case, the surface bends away from
the tangent plane partly towards one side of it, and partly towards
the other, and has thus, in some of its normal sections, curvatures
oppositely directed to those in others. In the latter case, the sur-
face on every side of the point bends away from the same side of
its tangent plane, and the curvatures of all normal sections are
similarly directed. Thus we may divide curved surfaces into Anti-
clastic and Synclastic. A saddle gives a good example of the
former class ; a ball of the latter. Curvatures in opposite directions,
with reference to the tangent plane, have of course different signs.
The outer portion of the surface of an anchor-ring is synclastic, the
inner anticlastic.
121. Meimier's Theorem. — The curvature of an oblique section
of a surface is equal to that of the normal section through the same
tangent line multiplied by the secant of the inclination of the planes
of the sections. This is evident from the most elementary con-
siderations regarding projections.
122. Elder's Theorem. — There are at every point of a synclastic
surface two normal sections, in one of which the curvature is a
maximum, in the other a minimum; and these are at right angles
to each other.
In an anticlastic surface there is maximum curvature (but in
opposite directions) in the two normal sections whose planes bisect
the angles between the lines in which the surface cuts its tangent
42 PRELIMINAR V.
plane. On account of the difference of sign, these may be con-
sidered as a maximum and a minimum.
Generally the sum of the curvatures at a point, in any two normal
planes at right angles to each other, is independent of the position
of these planes.
If - and - be the maximum and minimum curvatures at any
p a-
point, the curvature of a normal section making an angle 9 with the
normal section of maximum curvature is
- cos^ 6 + - sin^ 0,
P o-
which includes the above statements as particular cases.
123. Let F, p be two points of a surface indefinitely near to each
other, and let r be the radius of curvature of a normal section passing
through them. Then the radius of curvature of an oblique section
through the same points, inclined to the former at an angle a,
is r cos a (§ i2i). Also the length along the normal section, from
P to /, is less than that along the oblique section — since a given
chord cuts off an arc from a circle, longer the less is the radius
of that circle.
124. Hence, if the shortest possible line be drawn from one point
of a surface to another, its osculating plane, or plane of curvature,
is everywhere perpendicular to the surface.
Such a curve is called a Geodetic line. And it is easy to see that
it is the line in which a flexible and inextensible string would touch
the surface if stretched between those points, the surface being sup-
posed smooth.
125. A perfectly flexible but inextensible surface is suggested,
although not realized, by paper, thin sheet-metal, or cloth, when the
surface is plane ; and by sheaths of pods, seed-vessels, or the like,
when not capable of being stretched flat without tearing. The process
of changing the form of a surface by bending is called * developing.^
But the term ^ JDevelopable Surface' is commonly restricted to such
inextensible surfaces as can be developed into a plane, or, in com-
mon language, ' smoothed flat.'
126. The geometry or kinematics of this subject is a great contrast
to that of the flexible line (§ i6), and, in its merest elements, presents
ideas not very easily apprehended, and subjects of investigation that
have exercised, and perhaps even overtasked, the powers of some
of the greatest mathematicians.
127. Some care is required to form a correct conception of what
is a perfectly flexible inextensible surface. First let us consider a
plane sheet of paper. It is very flexible, and we can easily form
the conception from it of a sheet of ideal matter perfectly flexible.
KINEMATICS,
43
It is very inextensible ; that is to say, it yields very little to any
application of force tending to pull or stretch it in any direction,
up to the strongest it can bear without tearing. It does, of course,
stretch a little. It is easy to test that it stretches when under the
influence of force, and that it contracts again when the force is
removed, although not always to its original dimensions, as it
may and generally does remain to some sensible extent permanently
stretched. Also, flexure stretches one side and condenses the other
temporarily; and, to a less extent, permanently. Under elasticity we
may return to this. In the meantime, in considering illustrations of
our kinematical propositions, it is necessary to anticipate such phy-
sical circumstances.
128. The flexure of an inextensible surface which can be plane,
is a subject which has been well worked by geometrical investigators
and writers, and, in its elements at least, presents little difficulty. The
first elementary conception to be formed is, that such a surface (if
perfectly flexible), taken plane in the first place, may be bent about
any straight line ruled on it, so that the two plane parts may make
any angle with one another.
Such a line is called a 'generating line' of the surface to be
formed.
Next, we may bend one of these plane parts about any other line
which does not (within the limits of the sheet) intersect the former ;
and so on. If these lines are infinite in number, and the angles of
bending infinitely small, but such that their sum may be finite, we
have our plane surface bent into a curved surface, which is of course
'developable' (§ 125).
129. Lift a square of paper, free from folds, creases, or ragged
edges, gently by one corner, or otherwise, without crushing or forcing
it, or very gently by two points. It will hang in a form which is very
rigorously a developable surface; for although it is not absolutely
inextensible, yet the forces which tend to stretch or tear it, when it
is treated as above described, are small enough to produce absolutely
no sensible stretching. Indeed the greatest stretching it can expe-
rience without tearing, in any direction, is not such as can affect the
form of the surface much when sharp flexures, singular points, etc.,
are kept clear off".
130. Prisms and cylinders (when the lines of bending, § 128,
are parallel, and finite in number with finite angles, or infinite
in number with infinitely small angles), and pyramids and cones
(the lines of bending meeting in a point if produced), are clearly
included.
131. If the generating lines, or line-edges of the angles of bending,
are not parallel, they must meet, since they are in a plane when the
surface is plane. If they do not meet all in one point, they must
meet in several points : in general, each one meets its predecessor
and its successor in diff'erent points.
44
PRELIMINARY.
132. There Is still no difficulty in understanding the form of, say a
square, or circle, of the plane surface when bent as explained above,
provided it does not include any of these points
\ of intersection. When the number is infinite,
and the surface finitely curved, the developable
lines will, in general, be tangents to a curve (the
locus of the points of intersection when the
number is infinite). This curve is called the
edge of regression. The surface must clearly,
when complete (according to mathematical ideas),
consist of two sheets meeting in this edge of
regression (just as a cone consists of two
sheets meeting in the vertex), because each
tangent may be produced beyond the point
of contact, instead of stopping at it, as in the preceding diagram.
133. To construct a complete developable surface in two sheets
from its edge of regression — ■
Lay one piece of perfectly flat, un-
wrin'Kled, smooth-cut paper on the top
of another. Trace any curve on the
other, and let it have no point of in-
flection, but everywhere finite curvature.
Cut the paper quite away on the con-
cave side. If the curve traced is closed,
it must be cut open (see second diagram).
The limits to the extent that may be left uncut away, are the
tangents drawn outwards from the two ends, so that, in short, no
portion of the paper through which a real tangent does not pass
is to be left.
Attach the two sheets together by very slight paper or muslin
clamps gummed to them along the common curved edge. These
-7^ must be so slight as not to interfere sensibly with
>>•" the flexure of the two sheets. Take hold of one
corner of one sheet and lift the whole. The two
will open out into two sheets of a developable
surface, of which the curve, bending into a tor-
tuous curve, is the edge of regression. The tan-
gent to the curve drawn in one direction from
the point of contact, will always lie in one of the
sheets, and its continuation on the other side in the
other sheet. Of course a double-sheeted developable polyhedron can
be constructed by this process, by starting from a polygon instead
of a curve.
134. A flexible but perfectly inextensible surface, altered in form
in any way possible for it, must keep any hne traced on it un-
changed in length ; and hence any two intersecting lines unchanged
in mutual inclination. Hence, also, geodetic lines must remain
geodetic lines.
KINEMATICS. 45
135. We have now to consider the very important kinematical
conditions presented by the changes of volume or figure experienced
by a soHd or Hquid mass, or by a group of points whose positions
with regard to each other are subject to known conditions.
Any such definite alteration of form or dimensions is called a
Straiii.
Thus a rod which becomes longer or shorter is strained. Water,
when compressed, is strained. A stone, beam, or mass of metal, in a
building or in a piece of framework, if condensed or dilated in any
direction, or bent, twisted, or distorted in any way, is said to ex-
perience a strain. A ship is said to 'strain' if, in launching, or
when working in a heavy sea, the different parts of it experience
relative motions.
136. If, when the matter occupying any space is strained in any
way, all pairs of points of its substance which are initially at equal
distances from one another in parallel lines remain equidistant, it
may be at an altered distance ; and in parallel lines, altered, it may
be, from their initial direction ; the strain is said to be homogeneous.
137. Hence if any straight line be drawn through the body in its
initial state, the portion of the body cut by it will continue to be a
straight line when the body is homogeneously strained. For, if
ABC be any such line, AB and BC, being parallel to one line in the
initial, remain parallel to one line in the altered state; and therefore
remain in the same straight Hne with one another. Thus it follows
that a plane remains a plane, a parallelogram a parallelogram, and a
parallelepiped a parallelepiped.
138. Hence, also, similar figures, whether constituted by actual
portions of the substance, or mere geometrical surfaces, or straight or
curved lines passing through or joining certain portions or points of
the substance, similarly situated (i. e. having corresponding parameters
parallel) when altered according to the altered condition of the body,
remain similar and similarly situated among one another.
139. The lengths of parallel lines of the body remain in the same
proportion to one another, and hence all are altered in the same pro-
portion. Hence, and from § 137, we infer that any plane figure
becomes altered to another plane figure which is a diminished or
magnified orthographic projection of the first on some plane.
The elongation of the body along any line is the proportion which
the addition to the distance between any two points in that line bears
to their primitive distance.
140. Every orthogonal projection of an ellipse is an ellipse (the
case of a circle being included). Hence, and from § 139, we see
that an ellipse remains an ellipse; and an ellipsoid remains a sur-
face of which every plane section is an ellipse ; that is, remains an
ellipsoid.
46 PRELIMINAR Y.
141. The ellipsoid which any surface of the body initially spheri-
cal becomes in the altered condition, may, to avoid circumlocutions,
be called the Strain Ellipsoid.
142. In any absolutely unrestricted homogeneous strain there are
three directions (the three principal axes of the strain ellipsoid), at
right angles to one another, which remain at right angles to one
another in the altered condition of the body. Along one of these
the elongation is greater, and along another less, than along any
other direction in the body. Along the remaining one the elongation
is less than in any other line in the plane of itself and the first men-
tioned, and greater than along any other line in the plane of itself
and the second.
N'ote. — Contraction is to be reckoned as a negative elongation: the
maximum elongation of the preceding enunciation may be a mini-
mum contraction: the minimum elongation may be a maximum
contraction.
143. The ellipsoid Into which a sphere becomes altered may be
an ellipsoid of revolution, or, as it is called, a spheroid, prolate, or
oblate. There is thus a maximum or minimum elongation along
the axis, and equal minimum or maximum elongation along all lines
perpendicular to the axis.
Or it may be a sphere ; in which case the elongations are equal in
all directions. The effect is, in this case, merely an alteration of
dimensions without change of figure of any part.
144. The principal axes of a strain are the principal axes of the
ellipsoid into which it converts a sphere. The principal elongations
of a strain are the elongations in the direction of its principal axes.
145. When the positions of the principal axes, and the magnitudes
of the principal elongations of a strain are given, the elongation of
any line of the body, and the alteration of angle between any two
lines, may be obviously determined by a simple geometrical construc-
tion.
146. With the same data the alteration of angle between any tw^o
planes of the body may also be easily determined, geometrically.
147. Let the ellipse of the annexed diagram represent the section
of the strain ellipsoid through the greatest and least principal axes.
Let S'OS, TO The the two diameters of
this ellipse, which are equal to the mean
principal axis of the ellipsoid. Every
plane through O, perpendicular to the
plane of the diagram, cuts the ellipsoid
in an ellipse of which one principal axis
^' is the diameter in which it cuts the ellipse
of the diagram, and the other, the mean principal diameter of the
ellipsoid. Hence a plane through either SS' or TT', perpendicular
KINEMATICS,
47
to the plane of the diagram, cuts the ellipsoid in an ellipse of which
the two principal axes are equal, that is to say, in a circle. Hence
the elongations along all lines in either of these planes are equal to
the elongation along the mean principal axis of the strain ellipsoid.
148. The consideration of the circular sections of the strain ellip-
soid is highly instructive, and leads to important views with reference
to the analysis of the most general character of a strain. First let us
suppose there to be no alteration of volume on the whole, and neither
elongation nor contraction along the mean principal axis.
Let OX and OZ be the directions of maximum elongation and
maximum contraction respectively.
Let A be any point of the body
in its primitive condition, and A^ the
same point of the altered body, so
that OA=a.OA,
Now, if we take 0C= OA^^ and
if C^ be the position of that point
of the body which was in the
position C initially, we shall have
OC,^-OC, and therefore 0C =
'a '
OA. Hence the two triangles COA
and CpA^ are equal and similar.
Hence CA experiences no alteration of length, but takes the altered
position C^A^ in the altered position of the body. Similarly, if we
measure on XO produced, OA' and OA' equal respectively to OA
and OA^^ we find that the line CA' experiences no alteration in length,
but takes the altered position C^ A'^.
Consider now a plane of the body initially through CA perpen-
dicular to the plane of the diagram, which will be altered into a plane
through Ci^i, also perpendicular to the plane of the diagram. All
lines initially perpendicular to the plane of the diagram remain so,
and remain unaltered in length. ^C has just been proved to remain
unaltered in length. Hence (§ 139) all lines in the plane we have
just drawn remain unaltered in length and in mutual inclination.
Similarly we see that all lines in a plane through CA'^ perpendicular
to the plane of the diagram, altering to a plane through C-^A\^ per-
pendicular to the plane of the diagram, remain unaltered in length and
in mutual inclination.
149. The precise character of the strain we have now under con-
sideration will be elucidated by the following : — Produce CO, and take
OC and 0C\ respectively equal to OC and OC^. Join CA, C'A',
C\A^, and C\A\, by plain and dotted lines as in the diagram.
Then we see that the rhombus CA CA' (plain lines) of the body in
its initial state becomes the rhombus C\ ^1 C^ A'-^ (dotted) in the
altered condition. Now imagine the body thus strained to be
moved as a rigid body (i. e. with its state of strain kept unchanged)
48
PRELIMINARY.
J.;--...
till A^ coincides with A, and C\ with C\ keeping all the lines of
^ the diagram still in the same plane. A\Cx will
take a position in CA' produced, as shown in
the new diagram, and the original and the altered
parallelogram will be on the same base A C\ and
between the same parallels AC and CA\, and
their other sides will be equally inclined on the
two sides of a perpendicular to them. Hence,
irrespectively of any rotation, or other absolute
motion of the body not involving change of form
or dimensions, the strain under consideration
may be produced by holding fast and unaltered the plane of the
body through A C\ perpendicular to the plane of the diagram, and
making every plane parallel to it sHde, keeping the same distance,
through a space proportional to this distance (i.e. different planes
parallel to the fixed one slide through spaces proportional to their
distances).
150. This kind of strain is called a siinple shear. The plane of
a shear is a plane perpendicular to the undistorted planes, and
parallel to the lines of the relative motion. It has (i) the property
that one set of parallel planes remain each unaltered in itself ; (2)
that another set of parallel planes remain
each unaltered in itself. This other set is
got when the first set and the degree or
amount of shear are given, thus : — Let
CC^ be the motion of one point of one
plane, relative to a plane KL held fixed —
the diagram being in a plane of the shear.
Bisect CCx in N. Draw NA perpendicular
to it. A plane perpendicular to the plane
of the diagram, initially through AC^ and
finally through AC-^^ remains unaltered in
its dimensions.
151. One set of parallel undistorted planes and the amount of
their relative parallel shifting having been given, we have just seen
how to find the other set. The shear may be otherwise viewed, and
considered as a shifting of this second set of parallel planes, relative
to any one of them. The amount of this relative shifting is of course
equal to that of the first set, relatively to one of them.
152. The principal axes of a shear are the lines of maximum
elongation and of maximum contraction respectively. They may
be found from the preceding construction (§ 150), thus: — In the
plane of the shear bisect the obtuse and acute angles between the
planes destined not to become deformed. The former bisecting line
is the principal axis of elongation, and the latter is the principal
axis of contraction, in their initial positions. The former angle
(obtuse) becomes equal to the latter, its supplement (acute), in the
KINEMATICS. 49
altered condition of the body, and the ^-'-— ~-«^
lines bisecting the altered angles are the /""^ ^^^\
principal axes of the strain in the altered Dj^^^ 4^. --^B
Otherwise, taking a plane of shear for y ^^^---^^'^^j^^^^^ \|
the plane of the diagram, let AB be a Jj^ iT — ^~~j:j
line in which it is cut by one of either -"
set of parallel planes of no distortion. On any portion AB of this
as diameter, describe a semicircle. Through C, its middle point,
draw, by the preceding construction, CD the initial, and C£ the
final, position of an unstretched line. Join DA, DB, EA, EB.
DA, DB are the initial, and EA, EB the final, positions of the
principal axes,
153. The ratio of a shear is the ratio of elongation and contrac-
tion of its principal axes. Thus if one principal axis is elongated
in the ratio i : a, and the other therefore (§ 148) contracted in the
ratio a : I, a is called the ratio of the shear. It will be convenient
generally to reckon this as the ratio of elongation ; that is to say,
to make its numerical measure greater than unity.
In the diagram of § 152, the ratio of DB to EB, or of EA to DA,
is the ratio of the shear.
154. The amount of a shear is the amount of relative motion per
unit distance between planes of no distortion.
It is easily proved that this is equal to the excess of the ratio of
the shear above its reciprocal.
155. The planes of no distortion in a simple shear are clearly the
circular sections of the strain ellipsoid. In the ellipsoid of this
case, be it remembered, the mean axis remains unaltered, and is a
mean proportional between the greatest and the least axis.
156. If we now suppose all lines perpendicular to the plane of the
shear to be elongated or contracted in any proportion, without
altering lengths or angles in the plane of the shear, and if, lastly,
we suppose every line in the body to be elongated or contracted in
some other fixed ratio, we have clearly (§ 142) the most general
possible kind of strain.
157. Hence any strain whatever may be viewed as compounded
of a uniform dilatation in all directions, superimposed on a simple
elongation in the direction of one principal axis superimposed on a
simple shear in the plane of the two other principal axes,
158. It is clear that these three elementary component strains may
be applied in any other order as well as that stated. Thus, if the
simple elongation is made first, the body thus altered must get just
the same shear in planes perpendicular to the line of elongation
as the originally unaltered body gets when the order first stated is
followed. Or the dilatation may be first, then the elongation, and
finally the shear, and so on.
T. 4
so PRELIMINARY,
159. When the axes of the ellipsoid are lines of the body whose
direction does not change, the strain is said to be piire^ or unaccom-
panied by rotation. The strains we have already considered were
pure strains accompanied by rotations.
160. If a body experience a succession of strains, each unaccom-
panied by rotation, its resulting condition will generally be producible
by a strain and a rotation. From this follows the remarkable corol-
lary that three pure strains produced one after another, in any piece
of matter, each without rotation, may be so adjusted as to leave the
body unstrained, but rotated through some angle about some axis.
We shall have, later, most important and interesting applications to
fluid motion, which will be proved to be instantaneously, or differ-
entially, irrotational ; but which may result in leaving a whole fluid
mass merely turned round from its primitive position, as if it had
been a rigid body. [The following elementary geometrical in-
vestigation, though not bringing out a thoroughly comprehensive
view of the subject, affords a rigorous demonstration of the pro-
position, by proving it for a particular case.
Let us consider, as above (§ 150), a simple shearing motion. A
point O being held fixed, suppose the matter of the body in a plane,
cutting that of the diagram perpendicularly in CZ>, to move in this
plane from right to left parallel to CD ; and in other planes parallel
to it let there be motions proportional to their distances from O.
Consider first a shear from /* to P^\ then from P^ on to P^ ; and
let O be taken in a line through Pj, perpendicular to CD. During
the shear from P to /\
D a point Q moves of
course to Q^ through a
distance QQi = PPx-
Choose Q midway be-
tween P and /\, so that
P\Q.^ QiP=\P\P- Now, as we have seen above (§ 152), the
line of the body, which is the principal axis of contraction in the
shear from ^ to Q^^ is OA^ bisecting the angle QOE at the be-
ginning, and OA^, bisecting QyOE at the end, of the whole
motion considered. The angle between these two lines is half
the angle Q^OQ,', that is to say, is equal to P^OQ. Hence,
if the plane CD is rotated through an angle equal to PiOQ, in
the plane of the diagram, in the same way as the hands of a watch,
during the shear from Q to Q^^ or, which is the same thing, the
shear from Pto Pi, this shear will be effected without final rotation
of its principal axes. (Imagine the diagram turned round till OA^
lies along OA. The actual and the newly imagined position of CD
will show how this plane of the body has moved during such non-
rotational shear.)
Now, let the second step, P^ to P^, be made so as to complete
the whole shear, P to P2, which we have proposed to consider.
Such second partial shear may be made by the common shearing
kinematics: -^^
process parallel to the new position (imagined in the preceding
parenthesis) of CD^ and to make it also non-rotational, as its
-predecessor has been made, we must turn further round, in the
same direction, through an angle equal to QiOF^. Thus in these
two steps, each made non-rotational, we have turned the plane CD
round through an angle equal to QiOQ. But now, we have a whole
shear PF.2 ; and to make this as one non-rotational shear, we must
turn CD through an angle I^^ OF only, which is less than QiOQ by
the excess of P^OQ above QOP. Hence the resultant of the two
shears, PP^^ -^1^2' ^^^^"^ separately deprived of rotation, is a single
shear PP^, and a rotation of its principal axes, in the direction of
the hands of a watch, through an angle equal to QOP^ — POQ.
161. Make the two partial shears each non-rotationally. Return
from their resultant in a single non-rotational shear: we conclude with
the body unstrained, but turned through the angle QOP^-POQj in
the same direction as the hands of a watch.]
162. As there can be neither annihilation nor generation o^ matter
in any natural motion or action, the whole quantity of a fluid within
any space at any time must be equal to the quantity originally in
that space, increased by the whole quantity that has entered it, and
diminished by the whole quantity that has left it. This idea, when
expressed in a perfectly comprehensive manner for every portion of
a fluid in motion, constitutes what is commonly called the ' equation
of continuity.^
163. Two ways of proceeding to express this idea present
themselves, each aff"ording instructive views regarding the properties
of fluids. In one we consider a definite portion of the fluid ; follow
it in its motions; and declare that the average density of the substance
varies inversely as its volume. We thus obtain the equation ot con-
tinuity in an integral form.
The form under which the equation of continuity is most commonly
given, or the differential equation of continuity, as we may call it, ex-
presses that the rate of diminution of the density bears to the density,
at any instant, the same ratio as the rate of increase of the volume of
an infinitely small portion bears to the volume of this portion at the
same instant.
164. To find the differential equation of continuity, imagine a
space fixed in the interior of a fluid, and consider the fluid
which flows into this space, and the fluid which flows out of it,
across different parts of its bounding surface, in any time. If the
fluid is of the same density and incompressible, the whole quantity of
•matter in the space in question must remain constant at all times, and
therefore the quantity flowing in must be equal to the quantity flowing
out in any time. If, on the contrary, during any period of motion,
more fluid enters than leaves the fixed space, there will be condensa-
tion of matter in that space ; or if more fluid leaves than enters, there
will be dilatation. The rate of augmentation of the average density
$2 PRELIMINARY,
of the fluid, per unit of time, in the fixed space in question, bears to
the actual density, at any instant, the same ratio that the rate of
acquisition of matter into that space bears to the whole matter in that
space.
165. Several references have been made in preceding sections to
the number of independent variables in a displacement, or to the
degrees of freedom or constraint under which the displacement takes
place. It may be well, therefore, to take a general (but cursory) view
of this part of the subject itself.
166. A free point has //^r^^ degrees of freedom, inasmuch as the
most general displacement which it can take is resolvable into three,
parallel respectively to any three directions, and independent of each
other. It is generally convenient to choose these three directions of
resolution at right angles to one another.
If the point be constrained to remain always on a given surface,
one degree of constraint is introduced, or there are left but two
degrees of freedom. For we may take the normal to the surface
as one of three rectangular directions of resolution. No displacement
can be effected parallel to it : and the other two displacements, at
right angles to each other, in the tangent plane to the surface, are
independent.
If the point be constrained to remain on each of two surfaces, it
loses two degrees of freedom, and there is left but one. In fact,
it is constrained to remain on the curve which is common to both
surfaces, and along a curve there is at each point but one direction
of displacement.
167. Taking next the case of a free rigid system, we have evidently
six degrees of freedom to consider — three independent displacements
or translations in rectangular directions as a point has, and three
independent rotations about three mutually rectangular axes.
If it have one point fixed, the system loses three degrees of free-
dom ; in fact, it has now only the rotations above mentioned.
This fixed point may be, and in general is, a point of a continuous
surface of the body in contact with a continuous fixed surface. These
surfaces may be supposed * perfectly rough,' so that sliding may be
impossible.
If a second point be fixed, the body loses two more degrees of
freedom, and keeps only one freedom to rotate about the line joining
the two fixed points.
If a third point, not in a line with the other two, be fixed, the body
is fixed.
168. If one point of the rigid system is forced to remain on a
smooth surface, one degree of freedom is lost ; there remain yfz^^, two
displacements in the tangent plane to the surface, and three rotations.
As an additional degree of freedom is lost by each successive limita-
tion of a point in the body to a smooth surface, six such conditions
completely determine the position of the body. Thus if six points
KINEMATICS, 53
properly chosen on the barrel and stock of a rifle be made to rest on
six convex portions of the surface of a fixed rigid body, the rifle may
be replaced any number of times in precisely the same position, for
the purpose of testing its accuracy.
A fixed V under the barrel near the muzzle, and another under
the swell of the stock close in front of the trigger-guard, give four
of the contacts, bearing the weight of the rifle. A fifth (the one
to be broken by the recoil) is supplied by a nearly vertical fixed
plane close behind the second V, to be touched by the trigger-guard,
the rifle being pressed forward in its V's as far as this obstruction
allows it to go. This contact may be dispensed with and nothing
sensible of accuracy lost, by having a mark on the second V, and a
corresponding mark on barrel or stock, and sliding the barrel back-
wards or forwards in the V's till the two marks are, as nearly as
can be judged by eye, in the same plane perpendicular to the barrel's
axis. The sixth contact may be dispensed with by adjusting two
marks on the heel and toe of the butt to be as nearly as need be
in one vertical plane judged by aid of a plummet. This method
requires less of costly apparatus, and is no doubt more accurate and
trustworthy, and more quickly and easily executed, than the ordi-
nary method of clamping the rifle in a massive metal cradle set on a
heavy mechanical slide.
A geometrical clamp is a means of applying and maintaining
six mutual pressures between two bodies touching one another at
six points.
A 'geometrical slide* is any arrangement to apply five degrees
of constraint, and leave one degree of freedom, to the relative motion
of two rigid bodies by keeping them pressed together at just five
points of their surfaces.
Ex. I. The transit instrument would be an instance if one end
of one pivot, made slightly convex, were pressed against a fixed
vertical end-plate, by a spring pushing at the other end of the axis.
The other four guiding points are the points, or small areas, of con-
tact of the pivots on the Y*s.
Ex. 2. Let two rounded ends of legs of a three-legged stool
rest in a straight, smooth, V-shaped canal, and the third on a smooth
horizontal plane. Gravity maintains positive determinate pressures
on the five bearing points; and there is a determinate distribution
and amount of friction to be overcome, to produce the rectilineal
translational motion thus accurately provided for.
Ex. 3. Let only one of the feet rest in a V canal, and let
another rest in a trihedral hollow in line with the canal, the third
still resting on a horizontal plane. There are thus six bearing points,
one on the horizontal plane, two on the sides of the canal, and three
on the sides of the trihedral hollow : and the stool is fixed in a
determinate position as long as all these six contacts are unbroken.
Substitute for gravity a spring, or a screw and nut (of not infinitely
rigid material), binding the stool to the rigid body to which these
six planes belong. Thus we have a 'geometrical clamp,' which
54, PRELIMINARY.
clamps two bodies together with perfect firmness in a perfectly
definite position, without the aid of friction (except in the screw,
if a screw is used) ; and in various practical appUcations gives
very readily and conveniently a more securely firm connexion by
one screw slightly pressed, than a clamp such as those commonly
made hitherto by mechanicians can give with three strong screws
forced to the utmost.
Do away with the canal and let two feet (instead of only one) rest
on the plane, the other still resting in the conical hollow. The
number of contacts is thus reduced to five (three in the hollow
and two on the plane), and instead of a ' clamp ' we have again
a slide. This form of sHde, — a three-legged stool with two feet
resting on a plane and one in a hollow, — will be found very useful in
a large variety of applications, in which motion about an axis is de-
sired when a material axis is not conveniently attainable. Its first
application was to the 'azimuth mirror,' an instrument placed on
the glass cover of a mariner's compass and used for taking azimuths
of sun or stars to correct the compass, or of landmarks or other
terrestrial objects to find the ship's position. It has also been applied
to the ' Deflector,' an adjustible magnet laid on the glass of the
compass bowl and used, according to a principle first we believe
given by Sir Edward Sabine, to discover the 'semicircular' error
produced by the ship's iron. The movement may be made very
frictionless when the plane is horizontal, by weighting the move-
able body so that its centre of gravity is very nearly over the
foot that rests in the hollow. One or two guard feet, not to touch
the plane except in case of accident, ought to be added to give
a broad enough base for safety.
The geometrical slide and the geometrical clamp have both been
found very useful in electrometers, in the * siphon recorder,' and in
an instrument recently brought into use for automatic signalling
through submarine cables. An infinite variety of forms may be
given to the geometrical slide to suit varieties of application of the
general principle on which its definition is founded.
An old form of the geometrical clamp, with the six pressures pro-
duced by gravity, is the three V grooves on a stone slab bearing the
three legs of an astronomical or magnetic instrument. It is not
generally however so 'well-conditioned' as the trihedral hole, the
V groove, and the horizontal plane contact, described above.
There is much room for improvement by the introduction of
geometrical slides and geometrical clamps, in the mechanism of
mathematical, optical, geodetic, and astronomical instruments :.
which as made at present are remarkable for disregard of geome-
trical and dynamical principles in their slides, micrometer screws,
and clamps. Good workmanship cannot compensate for bad design,
whether in the safety-valve of an ironclad, or the movements and
adjustments of a theodolite.
169. If one point be constrained to remain in a curve, there remain
four degrees of freedom.
KINEMATICS,
55
If two points be constrained to remain in given curves, there are
four degrees of constraint, and we have left two degrees of freedom.
One of these may be regarded as being a simple rotation about the
line joining the constrained parts, a motion which, it is clear, the
body is free to receive. It may be shown that the other possible
motion is of the most general character for one degree of freedom ;
that is to say, translation and rotation in any fixed proportions, as of
the nut of a screw.
If one line of a rigid system be constrained to remain parallel to
itself, as for instance, if the body be a three-legged stool standing on
a perfectly smooth board fixed to a common window, shding in its
frame with perfect freedom, there remain three displacements and one
rotation.
But we need not farther pursue this subject, as the number of
combinations that might be considered is almost endless ; and those
already given suffice to show how simple is the determination of the
degrees of freedom or constraint in any case that may present itself.
170. One degree of constraint of the most general character, is not
producible by constraining one point of the body to a curve surface ;
but it consists in stopping one line of the body from longitudinal
motion, except accompanied by rotation round this line, in fixed
proportion to the longitudinal motion. Every other motion being
left unimpeded, there remains free rotation about any axis perpen-
dicular to that line (two degrees of freedom) ; and translation in any
direction perpendicular to the same line (two degrees of freedom).
These last four, with the one degree of freedom to screw, con-
stitute the five degrees of freedom, which, with one degree of con-
straint, make up the six elements. This condition is realized in the
following mechanical arrangement, which seems the simplest that
can be imagined for the purpose : —
Let a screw be cut on one shaft of a Hooke's joint, and let the
other shaft be joined to a fixed shaft by a second Hooke's joint.
A nut turning on that screw-shaft has the most general kind of
motion admitted when there is one degree of constraint. Or it is
subjected to just one degree of constraint of the most general cha-
racter. It has five degrees of freedom ; for it may move, ist, by
screwing on its shaft, the two Hooke's joints being at rest; 2nd,
it may rotate about either axis of the first Hooke's joint, or any axis
in their plane (two more degrees of freedom : being freedom to rotate
about two axes through one point) ; 3rd, it may, by the two Hooke's
joints, each bending, have translation without rotation in any direction
perpendicular to the link or shaft between the two Hooke's joints
(two more degrees of freedom). But it cannot have a motion of
translation parallel to the line of the link without a definite propor-
tion of rotation round this line ; nor can it have rotation round this
line without a definite proportion of translation parallel to it.
CHAPTER II.
DYNAMICAL LAWS AND PRINCIPLES.
; 171. In the preceding chapter we considered as a subject of pure
geometry the motion of points, lines, surfaces, and volumes, whether
taking place with or without change of dimensions and form ; and the
results we there arrived at are of course altogether independent of the
idea of matter^ and of \k\Q forces which matter exerts. We have here-
tofore assumed the existence merely of motion, distortion, etc.; we
now come to the consideration, not of how we might consider such
motion, etc., to be produced, but of the actual causes which in the
material world do produce them. The axioms of the present chapter
must therefore be considered to be due to actual experience, in the
shape either of observation or experiment. How such experience is
tp be conducted will form the subject of a subsequent chapter.
172. We cannot do better, at all events in commencing, than follow
Newton somewhat closely. Indeed the introduction to the Principia
contains in a most lucid form the general foundations of dynamics.
The Definitiofies and Axiomata, sive Leges Motus, there laid down,
require only a few amplifications and additional illustrations, suggested
by subsequent developments, to suit them to the present state of
science, and to make a much better introduction to dynamics than
we find in even some of the best modern treatises.
173. We cannot, of course, give a definition of Matter which will
satisfy the metaphysician ; but the naturalist may be content to know
matter as that which can he perceived by the senses ^ or as that which can
be acted upon by, or can exert, force. The latter, and indeed the
former also, of these definitions involves the idea of Force, which, in
point of fact, is a direct object of sense ; probably of all our senses,
and certainly of the * muscular sense.' To our chapter on Properties
of Matter we must refer for further discussion of the question, What
is matter!
174. The Quantity of Matter in a body, or, as we now call it, the
Mass of a body, is proportional, according to Newton, to the Volume
and the Density conjointly. In reality, the definition gives us the
meaning of density rather than of mass ; for it shows us that if twice
the original quantity of matter, air for example, be forced into a vessel
DYNAMICAL LA WS AND PRINCIPLES, 57
of given capacity, the density will be doubled, and so on. But it also
shows us that, of matter of uniform density, the mass or quantity is
proportional to the volume or space it occupies.
Let M be the mass, p the density, and V the volume, of a homo-
geneous body. Then
M=Fp;
if we so take our units that unit of mass is that of unit volume of a
body of unit density.
If the density be not uniform, the equation
M= Vfj
gives the Average (§ 26) density; or, as it is usually called, the Mean
density, of the body.
It is worthy of particular notice that, in this definition, Newton
says, if there be anything which freely pervades the interstices of all
bodies, this is not taken account of in estimating their Mass or
Density.
175. Newton further states, that a practical measure of the mass
of a body is its Weight. His experiments on pendulums, by which he
establishes this most important remark, will be described later, in our
chapter on Properties of Matter.
As will be presently explained, the unit mass most convenient for
British measurements is an imperial pound of matter.
176. The Qiiantity of Motion, or the Momentum^ of a rigid body
moving without rotation is proportional to its mass and velocity con-
jointly. The whole motion is the sum of the motions of its several
parts. Thus a doubled mass, or a doubled velocity, would correspond
to a double quantity of motion ; and so on.
Hence, if we take as unit of momentum the momentum of a unit
of matter moving with unit velocity, the momentum of a mass M
moving with velocity v is Mv.
Yll, Change of Quantity of Motion, or Change of Momentum, is
proportional to the mass moving and the change of its velocity
conjointly.
Change of velocity is to be understood in the general sense of § 31.
Thus, in the figure of that section, if a velocity represented by OA be
changed to another represented by OC, the change of velocity is
represented in magnitude and direction by A C
178. Pate of Change of Momentum, or Acceleration of Momentum, is
proportional to the mass moving and the acceleration of its velocity
conjointly. Thus (§ 44) the rate of change of momentum of a
felhng body is constant, and in the vertical direction. Again (§ 36)
the rate of change of momentum of a mass M, describing a circle of
MV^
fadius R, with uniform velocity V, is — 5— , and is directed to the
centre of the circle ; that is to say, it depends upon a change of di-
rection, not a change of speed, of the motion. ....
SS^ PRELIMINARY.
179. The Vis Viva, or Kinetic Energy, of a moving body is pro-
portional to the mass and the square of the velocity, conjointly. If
we adopt the same units of mass and velocity as before, there is
particular advantage in defining kinetic energy as halftht product of
the mass and the square of its velocity.
180. Rate of Change of Kinetic Energy (when defined as above) is
the product of the velocity into the component of acceleration of
momentum in the direction of motion.
Suppose the velocity of a mass M to be changed from v to v^ in
any time t j the rate at which the kinetic energy has changed is
-.\M{vf-'i^) = -M{:v-i),\{v^^v),
"^ow - M{v^-v) is the rate of change of momentum in the direc-
tion of motion, and J (v^ + v) is equal to v, if t be infinitely small.
Hence the above statement. It is often convenient to use Newton's
Fluxional notation for the rate of change of any quantity per unit of
time. In this notation (§ 28) v stands for - {v,~v) ; so that the rate
of change of^Mv^, the kinetic energy, is Mv . v. (See also §§229, 241.)
181. It is lo be observed that, in what precedes, with the exception of
the definition of density, we have taken no account of the dimensions
of the moving body. This is of no consequence so long as it does
not rotate, and so long as its parts preserve the same relative positions
amongst one another. In this case we may suppose the whole of the
matter in it to be condensed in one point or particle. We thus speak
of a material particle, as distinguished from 2^ geometrical poiftt. If the
body rotate, or if its parts change their relative positions, then we
cannot choose any one point by whose motions alone we may de-
termine those of the other points. In such cases the momentum and
change of momentum of the whole body in any direction are, the
sums of the momenta, and of the changes of momentum, of its parts,
in these directions ; while the kinetic energy of the whole, being non-
directional, is simply the sum of the kinetic energies of the several
parts or particles.
182. Matter has an innate power of resisting external influences,
so that every body, so far as it can, remains at rest, or moves uni-
formly in a straight Hne.
This, the Inertia of matter, is proportional to the quantity of matter
in the body. And it follows that some cause is requisite to disturb a
body's uniformity of motion, or to change its direction from the
natural rectilinear path.
183. Eorce is any cause which tends to alter a body's natural state
of rest, or of uniform motion in a straight line.
Force is wholly expended in the Action it produces ; and the body,
after the force ceases to act, retains by its inertia the direction of
DYNAMICAL LA WS AND PRINCIPLES. 59
motion and the velocity which were given to it. Force may be of
divers kinds, as pressure, or gravity, or friction, or any of the attractive
or repulsive actions of electricity, magnetism, etc.
184. The three elements specifying a force, or the three elements
which must be known, before a clear notion of the force under con-
sideration can be formed, are, its place of application, its direction,
and its magnitude.
^ {a) The place of application of a force. The first case to be con-
sidered is that in which the place of application is a point. It has
been shown already in what sense the term ' point ' is to be taken,
and, therefore, in what way a force may be imagined as acting at a
point. In reality, however, the place of application of a force is
always either a surface or a space of three dimensions occupied by
matter. The point of the finest needle, or the edge of the sharpest
knife, is still a surface, and acts as such on the bodies to which it
may be applied. Even the most rigid substances, when brought
together, do not touch at a point merely, but mould each other so
as to produce a surface of application. On the other hand, gravity
is a force of which the place of application is the whole matter of the
body whose weight is considered ; and the smallest particle of matter
that has weight occupies some finite portion of space. Thus it is to
be remarked, that there are two kinds of force, distinguishable by
their place of application — force whose place of application is a
surface, and force whose place of application is a solid. When a
heavy body rests on the ground, or on a table, force of the second
character, acting downwards, is balanced by force of the first character
acting upwards.
{b) The second element in the specification of a force is its
direction. The direction of a force is the line in which it acts.
If the place of application of a force be regarded as a point, a
line through that point, in the direction in which the force tends to
move the body, is the direction of the force. In the case of a force
distributed over a surface, it is frequently possible and convenient
to assume a single point and a single line, such that a certain force
acting at that point in that Une would produce the same effect as is
really produced.
{c) The third element in the specification of a force is its magnitude.
This involves a consideration of the method followed in dynamics for
measuring forces. Before measuring anything it is necessary to have
a unit of measurement, or a standard to which to refer, and a prin-
ciple of numerical specification, or a mode of referring to the standard.
These will be supplied presently. See also § 224, below.
185. The Measure of a Force is the quantity of motion which it
produces in unit of time.
The reader, who has been accustomed to speak of a force of so
many pounds, or so many tons, may be reasonably startled when he
finds that Newton gives no countenance to such expressions. The
method is not correct unless it be specified at what part of the earth's
66^ PRELIMINARY,
surface the pound, or other definite quantity of matter named, is to
be weighed ; for the weight of a given quantity of matter differs in
different latitudes.
It is often, however, convenient to use instead of the absolute
unit (§ i88), the gravitation unit — which is simply the weight of unit
mass. It must, of course, be specified in what latitude the observation
is made. Thus, let W be the mass of a body in pounds; g the
velocity it would acquire in falling for a second under the influence
of its weight, or the earth's attraction diminished by centrifugal
force ; and P its weight measured in kinetic or absolute units. We
have p- j^g^
The force of gravity on the body, in gravitation units, is W.
186. According to the system commonly followed in mathe-
matical treatises on dynamics till fourteen years ago, when a small
instalment of the first edition of the present work was issued for
the use of our students, the unit of mass was g times the mass of
the standard or unit weight. This definition, giving a varying and a
very unnatural unit of mass, was exceedingly inconvenient. By taking.
the gravity of a constant mass for the unit of force it makes the unit
of force greater in high than in low latitudes. In reality, standards
of weight are masses^ not forces. They are employed primarily in
commerce for the purpose of measuring out a definite quantity of
matter; not an amount of matter which shall be attracted by the
earth with a given force.
A merchant, with a balance and a set of standard weights, would
give his customers the same quantity of the same kind of matter
however the earth's attraction might vary, depending as he does upon
weights for his measurement ; another, using a spring-balance, would
defraud his customers in high latitudes, and himself in low, if his
instrument (which depends on constant forces and not on the gravity
of constant masses) were correctly adjusted in London.
It is a secondary application of our standards of weight to employ
them for the measurement oi forces, such as steam pressures, mus-
cular power, etc. In all cases where great accuracy is required,
the results obtained by such a method have to be reduced to
what they would have been if the measurements of force had been
made by means of a perfect spring-balance, graduated so as to
indicate the forces of gravity on the standard weights in some con-
ventional locality.
It is therefore very much simpler and better to take the imperial
pound, or other national or international standard weight, as, for
instance, the gramme (see the chapter on Measures and Instru-
ments), as the unit of mass, and to derive from it, according to
Newton's definition above, the unit of force. This is the method
which Gauss has adopted in his great improvement dl the system of
measurement of forces. ^x
187. The formula, deduced by Clairault from observation, and a
certain theory regarding the figure and density of the earth, may be
DYNAMICAL LAWS AND PRINCIPLES. €i
'employed to calculate the most probable value of the apparent force
of gravity, being the resultant of true gravitation and centrifugal force,
in any locality where no pendulum observation of sufficient accuracy
has been made. This formula, with the two coefficients which it
involves, corrected according to modern pendulum observations, is
as follows : —
Let G be the apparent force of gravity on a unit mass at the
equator, and g that in any latitude X ; then
^=6^(i + -oo5i3sin'X).
The value of G^ in terms of the absolute unit, to be explained
immediately, is
32*088.
According to this formula, therefore, polar gravity will be
g= 32-088 X 1-00513 = 32-252.
188. As gravity does not furnish a definite standard, independent
of locality, recourse must be had to something else. The principle
of measurement indicated as above by Newton, but first introduced
practically by Gauss in connexion with national standard masses,
furnishes us with what we want. According to this principle, the
standard or unit force is that force which^ acti?ig oft a natioftal standard
unit of matter during the unit of time, generates the unit of velocity.
This is known as Gauss' absolute unit ; absolute, because it fur-
nishes a standard force independent of the differing amounts of
gravity at different localities. It is however terrestrial and incon-
stant if the unit of time depends on the earth's rotation, as it does
in our present system of chronometry. The period of vibration of
a piece of quartz crystal of specified shape and size and at a stated
temperature (a tuning-fork, or bar, as one of the bars of glass used
in the ' musical glasses ') gives us a unit of time which is constant
through all space and all time, and independent of the earth. A
unit of force founded on such a unit of time would be better entitled
to the designation absolute than is the * absolute unit ' now generally
adopted, which is founded on the mean solar second. But this de-
pends essentially on one particular piece of matter, and is therefore
liable to all the accidents, etc. which affect so-called National
Standards however carefully they may be preserved, as well as to
the almost insuperable practical difficulties which are experienced
when we attempt to make exact copies of them. Still, in the present
state of science, we are really confined to such approximations. The
recent discoveries due to the Kinetic theory of gases and to Spectrum
analysis (especially when it is applied to the light of the heavenly
bodies) indicate to us natural standard pieces of matter such as
atoms of hydrogen, or sodium, ready made in infinite numbers, all
absolutely alike in every physical property. The time of vibration
of a sodium particle corresponding to any one of its modes of vibra-
tion, is known to be absolutely independent of its position in the
universe, and it will probably remain the same so long as the particle
52 PRELIMINARY,
itself exists. The wave-length for that particular ray, i.e. the space
through which light is propagated in vacuo during the time of one
complete vibration of this period, gives a perfectly invariable unit of
length ; and it is possible that at some not very distant day the mass
of such a sodium particle may be employed as a natural standard for
the remaining fundamental unit. This, the latest improvement made
upon our original suggestion of a Feretmial Springy is due to Clerk
Maxwell.
189. The absolute unit depends on the unit of matter, the unit of
time, and the unit of velocity; and as the unit of velocity depends on
the unit of space and the unit of time, there is, in the definition, a
single reference to mass and space, but a double reference to time ;
and this is a point that must be particularly attended to.
190. The unit of mass may be the British imperial pound, or,
better, the gramme : the unit of space the British standard foot, or,
better, the centimetre ; and the unit of time the mean solar second.
We accordingly define the British absolute unit force as ' the force
which, acting on one pound of matter for one second, generates a
velocity of one foot per second.'
191. To render this standard intelligible, all that has to be done is
to find how many absolute units will produce, in any particular locality,
the same effect as the force of gravity on a given mass. The way to
do this is to measure the effect of gravity in producing acceleration
on a body unresisted in any way. The most accurate method is
indirect, by means of the pendulum. The result of pendulum ex-
periments made at Leith Fort, by Captain Kater, is, that the velocity
acquired by a body falling unresisted for one second is at that place
32*207 feet per second. The preceding formula gives exactly 32*2,
for the latitude 55° 35', which is approximately that of Edinburgh.
The variation in the force of gravity for one degree of difference of
latitude about the latitude of Edinburgh is only '0000832 of its own
amount. It is nearly the same, though somewhat more, for every
degree of latitude southwards, as far as the southern limits of the
British Isles. On the other hand, the variation per degree would be
sensibly less, as far north as the Orkney and Shetland Isles. Hence
the augmentation of gravity per degree from south to north through-
out the British Isles is at most about -^^kw^ ^^ its whole amount in
any locality. The average for the whole of Great Britain and Ireland
differs certainly but little from 32-2. Our present application is, that
the force of gravity at Edinburgh is 32*2 times the force which, acting
on a pound for a second, would generate a velocity of one foot per
second; in other words, 32*2 is the number of absolute units which
measures the weight of a pound in this latitude. Thus, speaking
very roughly, the British absolute unit of force is equal to the weight
of about half an ounce.
192. Forces (since they involve only direction and magnitude) may
be represented, as velocities are, by straight lines in their directions,
and of lengths proportional to their magnitudes, respectively.
DYNAMICAL LAWS AND PRINCIPLES. (i^
Also the laws of composition and resolution of any number of
forces acting at the same point, are, as we shall shov/ later (§ 221),
the same as those which we have already proved to hold for velo-
cities; so that with the substitution of force for velocity, §§ 30, 31
are still true.
193. The Component of a force in any direction, sometimes
called the Effective Component in that direction, is therefore found
by multiplying the magnitude of the force by the cosine of the
angle between the directions of the force and the component. The
remaining component in this case is perpendicular to the other.
It is very generally convenient to resolve forces into components
parallel to three lines at right angles to each other; each such reso-
lution being effected by multiplying by the cosine of the angle
concerned.
194. [If any number of points be placed in any positions in space,
another can be found, such that its distance from any plane what-
ever is the mean of their distances from that plane ; and if one or
more of the given points be in motion, the velocity of the mean
point perpendicular to the plane is the mean of the velocities of
the others in the same direction.
If we take two points A^, A2, the middle point, P^, of the line
joining them is obviously distant from any plane whatever by a
quantity equal to the mean (in this case the half sum or difference
as they are on the same or on opposite sides) of their distances
from that plane. Hence tzaice the distance of P2 from any plane
is equal to the (algebraic) sum of the distances of A-^, A from it.
Introducing a third point A^, if we join A^P^ and divide it in P^
so that A-iP^- 2P^P2, three times the distance of P^ from any plane
is equal to the sum of the distance of A^ and twice that of P^ from
the same plane: i. e. to the sum of the distances of A^, A. 2, and A^
from it ; or its distance is the mean of theirs. And so on for any
number of points. The proof is exceedingly simple. Thus suppose
Pn to be the mean of the first ;/ points A^, A^^.-.A^^; and A^^^ any
other point. Divide A^^^P^ in P^^^ so that A^^^P^^^ = nP^
Then from P^, -^„+i> ^„+i> draw perpen-
diculars to any plane, meeting it in 6*, 7] V.
Draw P^QR parallel to STV, Then
QK.. : RA^, :: PP ,, : PA^,, :: 1 : n + 1,
Hence n + iQP^+^ = RA^^^. Add to these
71+ I ^7" and its equal nP^S+RV, and we get
'^^^{Q^n..^QT)-nP^S + RV+RA^,,,
i.e. n+iP^,,T=nP,S + A„^^V,
In words, n+ 1 times the distance of P^^^ from any plane is equal
to that of ^^+1 with n times that of P^, i. e. equal to the sum of the
64 PRELIMINARY.
distances of A^^ ^^j-'-^n+i ^^^m the plane. Thus if the proposition
be true for any number of points, it is true for one more — and so on
- — but it is obviously true for two, hence for three, and therefore
generally. And it is obvious that the order in which the points are
taken is immaterial.
As the distance of this point from any plane is the mean of the
distances of the given ones, the rate of increase of that distance,
i. e. the velocity perpendicular to the plane, must be the mean of the
rates of increase of their distances — i. e. the mean of their velocities
perpendicular to the plane.]
195. The Centre of Inertia or Mass of a system of equal material
points (whether connected with one another or not) is the point
whose distance is equal to their average distance from any plane
whatever (§ 194).
A group of material points of unequal masses may always be
imagined as composed of a greater number of equal material points,
because we may imagine the given material points divided into dif-
ferent numbers of very small parts. In any case in which the magni-
tudes of the given masses are incommensurable, we may approach as
near as we please to a rigorous fulfilment of the preceding statement,
by making the parts into which we divide them sufficiently small.
On this understanding the preceding definition may be applied
to define the centre of inertia of a system of material points, whether
given equal or not. The result is equivalent to this : —
The centre of inertia of any system of material points whatever
(whether rigidly connected with one another, or connected in any
way, or quite detached), is a point whose distance from any plane
is equal to the sum of the products of each mass into its distance
from the same plane divided by the sum of the masses.
We also see, from the proposition stated above, that a point whose
distance from three rectangular planes fulfils this condition, must
fulfil this condition also for every other plane.
The co-ordinates of the centre of inertia, of masses Wj, Wg, etc.,
at points {x^y jFi, ^1), (^2) y^^ ^i)i ^tc, are given by the following
formulae : —
_ _ w-^x^ + Te/jy^a + Gtc. _ '^wx _ _ Srqy - _ ^7vz
~ W1 + W2 + etc. ~^w ' ^w * ^w '
These formulae are perfectly general, and can easily be put into
the particular shape required for any given case.
The Centre of Inertia or Mass is thus a perfectly definite point in
every body, or group of bodies. The term Centre of Gravity is often
very inconveniently used for it. The theory of the resultant action of
gravity, which will be given under Abstract Dynamics, shows that,
except in a definite class of distributions of matter, there is no fixed
point which can properly be called the Centre of Gravity of a rigid
body. In ordinary cases of terrestrial gravitation, however, an ap-
proximate solution is available, according to which, in common par-
lance, the term Centre of Gravity may be used as equivalent to
DYNAMICAL LAWS AND PRINCIPLES. 65
Centre of Inertia ; but it must be carefully remembered that the fun-
damental ideas involved in the two definitions are essentially different.
The second proposition in § 194 may now evidently be stated
thus : — The sum of the momenta of the parts of the system in any
direction is equal to the momentum in the same direction of a mass
equal to the sum of the masses moving with a velocity equal to the
velocity of the centre of inertia.
196. The mean of the squares of the distances of the centre of
p inertia, /, from each of the points of a system
^^ is less than the mean of the squares of the dis-
^yy i tance of any other point, O, from them by the
^^^ / \ square of 01. Hence the centre of inertia is
^^ y j^ the point the sum of the squares of whose
^ i U distances from any given points is a minimum.
For OP" = or + IP' + 2OIIQ, P being any one of the points
and PQ perpendicular to 01. But IQ is the distance of P from
a plane through / perpendicular to OQ. Hence the mean of all
distances, /ft is zero. Hence
(mean of IP') = (mean of OP') - 0I\ which is the proposition.
197. Again, the mean of the squares of the distances of the points
of the system from any line, exceeds the corresponding quantity for
a parallel line through the centre of inertia, by the square of the
distance between these lines.
For in the above figure, let the plane of the paper represent a
plane through / perpendicular to these lines, O the point in which
the first line meets it, P the point in which it is met by a parallel
line through any one of the points of the system. Draw, as before,
PQ perpendicular to 01. Then PI is the perpendicular distance,
from the axis through /, of the point of the system considered, PO
is its distance from the first axis, 01 the distance between the two
axes.
Then, as before,
(mean of OP') = OP + (mean of IP')\
since the mean of IQ is still zero, IQ being the distance of a
point of the system from the plane through / perpendicular to 01.
198. If the masses of the points be unequal, it is easy to see (as
in § 195) that the first of these theorems becomes —
The sum of the squares of the distances of the parts of a system
from any point, each multiplied by the mass of that part, exceeds the
corresponding quantity for the centre of inertia by the product of
the square of the distance of the point from the centre of inertia, by
the whole mass of the system.
Also, . the sum of the products of the mass of each part of
a system by the square of its distance from any axis is called the
Moment of Inertia of the system about this axis ; and the second
proposition above is equivalent to —
66 PRELIMINARY,
The moment of inertia of a system about any axis is equal to the
moment of inertia about a parallel axis through the centre of inertia,
/, together with the moment of inertia, about the first axis, of the
whole mass supposed condensed at /.
199. The Moment of any physical agency is the numerical mea-
sure of its importance. Thus, the moment of inertia of a body
round an axis (§ 198) means the importance of its inertia relatively
to rotation round that axis. Again, the moment of a force round a
point or round a line (§ 46), signifies the measure of its importance as
regards producing or balancing rotation round that point or round
that line.
It is often convenient to represent the moment of a force by a line
numerically equal to it, drawn through the vertex of the triangle
representing its magnitude, perpendicular to its plane, through the
front of a watch held in the plane with its centre at the point, and
facing so that the force tends to turn round this point in a direction
opposite to the hands. The moment of a force round any axis is the
moment of its component in any plane perpendicular to the axis,
round the point in which the plane is cut by the axis. Here we
imagine the force resolved into two components, one parallel to the
axis, which is ineffective so far as rotation round the axis is con-
cerned; the other perpendicular to the axis (that is to say, having its
line in any plane perpendicular to the axis). This latter component
may be called the effective component of the force, with reference
to rotation round the axis. And its moment round the axis may be
defined as its moment round the nearest point of the axis, which is
equivalent to the preceding definition. It is clear that the moment
of a force round any axis, is equal to the area of the projection on
any plane perpendicular to the axis, of the figure rejoresenting its
moment round any point of the axis.
200. [The projection of an area, plane or curved, on any plane,
is the area included in the projection of its bounding line.
If we imagine an area divided into any number of parts, the pro-
jections of these parts on any plane make up the projection of the
whole. But in this statement it must be understood that the areas
of partial projections are to be reckoned as positive if particular
sides, which, for brevity, we may call the outside of the projected
area and the front of the plane of projection, face the same way,
and negative if they face oppositely.
Of course if the projected surface, or any part of it, be a plane area
at right angles to the plane of projection, the projection vanishes.
The projections of any two shells having a common edge, on any
plane, are equal. The projection of a closed surface (or a shell with
evanescent edge), on any plane, is nothing.
Equal areas in one plane, or in parallel planes, have equal projec-
tions on any plane, whatever may be their figures.
Hence the projection of any plane figure, or of any shell edged
by a plane figure, on another plane, is equal to its area, multiplied
DYNAMICAL LAWS AND PRINCIPLES. 67
by the cosine of the angle at which its plane is inclined to the plane
of projection. This angle is acute or obtuse, according as the out-
side of the projected area, and the front of the plane of projection,
face on the whole towards the same parts, or oppositely. Hence
lines representing, as above described, moments about a point in
different planes, are to be compounded as forces are. See an
analogous theorem in § 107.]
201. A Couple is a pair of equal forces acting in dissimilar direc-
tions in parallel lines. The Motnmt of a couple is the sum of the
moments of its forces about any point in their plane, and is therefore
equal to the product of either force into the shortest distance between
their directions. This distance is called the Ann of the couple.
The Axis of a Couple is a line drawn from any chosen point of
reference perpendicular to the plane of the couple, of such magnitude
and in such direction as to represent the magnitude of the moment,
and to indicate the direction in which the couple tends to turn. The
most convenient rule for fulfilling the latter condition is this: — Hold
a watch with its centre at the point of reference, and with its plane
parallel to the plane of the couple. Then, according as the motion
of the hands is contrary to, or along with the direction in which the
couple tends to turn, draw the axis of the couple through the face or
through the back of the watch. It will be found that a couple is
completely represented by its axis, and that couples are to be resolved
and compounded by the same geometrical constructions performed
with reference to their axes as forces or velocities, with reference to
the lines directly representing them.
202. By introducing in the definition of moment of velocity (§ 46)
the mass of the moving body as .a factor, we have an important
element of dynamical science, the Moment of Mometiium. The
laws of composition and resolution are the same as those already
explained.
203. [If the point of application of a force be displaced through
a small space, the resolved part of the displacement in the direction
of the force has been called its Virtual Velocity. This is positive or
negative according as the virtual velocity is in the same, or in the
opposite, direction to that of the force.
The product of the force, into the virtual velocity of its point of
application, has been called the Virtual Moment of the force. These
terms we have introduced since they stand in the history and develop-
ments of the science ; but, as we shall show further on, they are
inferior substitutes for a far more useful set of ideas clearly laid down
by Newton.]
204. A force is said to do ivork if its place of application has a
positive component motion in its direction ; and the work done by it
is measured by the product of its amount into this component motion.
Generally, unit of work is done by unit force acting through unit
space. In lifting coals from a pit, the amount of work done is
5—2
68 PRELIMINARY.
proportional to the weight of the coals lifted; that is, to the force
overcome in raising them ; and also to the height through which they
are raised. The unit for the measurement of work adopted in practice
by British engineers, is that required to overcome a force equal to the
weight of a pound through the space of a foot ; and is called a Foot-
Powid. (See § 185.)
In purely scientific measurements, the unit of work is not the foot-
pound, but the kinetic unit force (§ 190) acting through unit of space.
Thus, for example, as we shall show further on, this unit is adopted
in measuring the work done by an electric current, the units for
electric and magnetic measurements being founded upon the kinetic
unit force.
If the weight be raised obliquely, as, for instance, along a smooth
inclined plane, the space through which the force has to be overcome
is increased in the ratio of the length to the height of the plane ; but
the force to be overcome is not the whole weight, but only the resolved
part of the weight parallel to the plane; and this is less than the
weight in the ratio of the height of the plane to its length. By multi-
plying these two expressions together, we find, as we might expect,
that the amount of work required is unchanged by the substitution of
the oblique for the vertical path.
205. Generally, for any force, the work done during an indefinitely
small displacement of the point of application is the virtual moment
of the force (§ 203), or is the product of the resolved part of the force
in the direction of the displacement into the displacement.
From this it appears, that if the motion of the point of application
be always perpendicular to the direction in which a force acts, such a
force does no work. Thus the mutual normal pressure between a
fixed and moving body, the tension of the cord to which a pendulum
bob is attached, or the attraction of the sun on a planet if the planet
describe a circle with the sun in the centre, are all instances in which
no work is done by the force.
206. The work done by a force, or by a couple, upon a body
turning about an axis, is the product of the moment of either into the
angle (in circular measure) through which the body acted on turns, if
the moment remains the same in all positions of the body. If the
moment be variable, the above assertion is only true for indefinitely
small displacements, but maybe made accurate by employing the proper
average moment of the force or of the couple. The proof is obvious.
207. Work done on a body by a force is always shown by a cor-
responding increase of vis viva, or kinetic energy, if no other forces
act on the body which can do work or have work done against them.
If work be done against any forces, the increase of kinetic energy is
less than in the former case by the amount of work so done. In
virtue of this, however, the body possesses an equivalent in the form
of Fofe?itial Efiergy (§ 239), if its physical conditions are such that
these forces will act equally, and in the same directions, if the motion
of the system is reversed. Thus there may be no change of kinetic
DYNAMICAL LAWS AND PRINCIPLES. 69
energy produced, and the work done may be wholly stored up as
potential energy.
Thus a weight requires work to raise it to a height, a spring requires
work to bend it, air requires work to compress it, etc. ; but a raised
weight, a bent spring, compressed air, etc., are stores of energy which
can be made use of at pleasure.
208. In what precedes we have given some of Newton's Definitiones
nearly in his own words ; others have been enunciated in a form more
suitable to modern methods ; and some terms have been introduced
which were invented subsequent to the publication of the Principia.
But the Axiomata^ sive Leges Mofih, to which we now proceed, are
given in Newton's own words. The two centuries which have nearly
elapsed since he first gave them have not shown a necessity for any
addition or modification. The first two, indeed, were discovered by
Galileo : and the third, in some of its many forms, was known to
Hooke, Huyghens, Wallis, Wren, and others, before the publication
of the Principia. Of late there has been a tendency to divide the
second law into two, called respectively the second and third, and to
ignore the third entirely, though using it directly in every dynamical
problem ; but all who have done so have been forced indirectly to
acknowledge the incompleteness of their substitute for Newton's system,
by introducing as an axiom what is called D'Alembert's principle, which
is really a deduction from Newton's rejected third law. Newton's own
interpretation of his third law directly points out not only D'Alembert's
principle, but also the modern principles of Work and Energy.
209. An Axiom is a proposition, the truth of which must be ad-
mitted as soon as the terms in which it is expressed are clearly
understood. And, as we shall show in our chapter on ' Experience,'
physical axioms are axiomatic to those who have sufficient knowledge
of physical phenomena to enable them to understand perfectly what
is asserted by them. Without further remark we shall give Newton^s
Three Laws ; it being remembered that, as the properties of matter
might have been such as to render a totally different set of laws
axiomatic, these laws must be considered as resting on convictions
drawn from observation and experiment, not on intuitive perception.
210. Lex I. Corpus omne perseuerare in statu suo qtdescendi vel
movendi uniformiter i?t directimi^ nisi qtiatenus illud d viribus impressis
cogitur statum suum niutare.
Every body cojitinues in its state of rest or of uniform motion in a
straight line, except in so far as it may be compelled by impressed forces
to change that state.
211. The meaning of the term Rest, in physical science, cannot be
absolutely defined, inasmuch as absolute rest nowhere exists in nature.
If the universe of matter were finite, its centre of inertia might fairly
be considered as absolutely at rest ; or it might be imagined to be
moving with any uniform velocity in any direction whatever through
infinite space. But it is remarkable that the first law of motion
70 PRELIMINARY.
enables us (§215, below) to explain what may be called directional
rest. Also, as will be seen farther on, a perfectly smooth spherical
body, made up of concentric shells, each of uniform material and
density throughout, if made to revolve about an axis, will, /// spite of
impressed forces, revolve with uniform angular velocity, and will main-
tain its axis of revolution in an absolutely fixed direction. Or, as will
soon be shown (§ 233), the plane in which the moment of momentum
of the universe (if finite) round its centre of inertia is the greatest,
which is clearly determinable from the actual motions at any instant,
is fixed in direction in space.
212. We may logically convert the assertion of the first law of
motion as to velocity into the following statements : —
The times during which any particular body, not compelled by
force to alter the speed of its motion, passes through equal spaces,
are equal. And, again — Every other body in the universe, not com-
pelled by force to alter the speed of its motion, moves over equal
spaces in successive intervals, during which the particular chosen body
moves over equal spaces.
213. The first part merely expresses the convention universally
adopted for the measurement of Time. The earth in its rotation
about its axis, presents us with a case of motion in which the con-
dition of not being compelled by force to alter its speed, is more
nearly fulfilled than in any other which we can easily or accurately
observe. And the numerical measurement of time practically rests
on defining equal intervals of time, as times duri?ig which the earth turns
through equal angles. This is, of course, a mere convention, and
not a law of nature; and, as we now see it, is a part of Newton's
first law.
214. The remainder of the law is not a convention, but a great
truth of nature, which we may illustrate by referring to small and
trivial cases as well as to the grandest phenomena we can conceive.
A curling-stone, projected along a horizontal surface of ice, travels
equal distances, except in so far as it is retarded by friction and by
the resistance of the air, in successive intervals of time during which
the earth turns through equal angles. The sun moves through equal
portions of interstellar space in times during which the earth turns
through equal angles, except in so far as the resistance of interstellar
matter, and the attraction of other bodies in the universe, alter his
speed and that of the earth's rotation.
215. If two material points be projected from one position. A, at
the same instant with any velocities in any directions, and each left to
move uninfluenced by force, the line joining them will be always
parallel to a fixed direction. For the law asserts, as we have seen,
that AP : AP' :: AQ : AQ\ i( P, Q, and again P\ Q', are simulta-
neous positions ; and therefore P(2 is parallel to P'Q'. Hence if four
material points O, P, Q, R are all projected at one instant from one
position, OP, OQ, OR are fixed directions of reference ever after.
D YNAMICAL LA WS AND PRINCIPLES. 7 1
But, practically, the determination of fixed directions in space
(§ 233) is made to depend upon the rotation of groups of particles
exerting forces on each other, and thus involves the Third Law of
Motion.
216. The whole law is singularly at variance with the tenets of the
ancient philosophers, who maintained that circular motion is perfect.
The last clause, '■nisi quate7ius^ etc., admirably prepares for the
introduction of the second law, by conveying the idea that it is force
alo7ie ivhich can produce a change of motion. How, we naturally in-
quire, does the change of motion produced depend on the magnitude
and direction of the force which produces it? The answer is —
217. Lex II. Mutatio7iem mot us proportion akm esse vi 7notrici i7n'
pressae, et fieri secundum Ii7iea77i recta77i qua vis ilia imprimitur.
Cha7ige of 77iotio7i is proportiojialto the i77ipressed force^ and takes place
in the di7'ection of the straight line in which thefo7'ce acts.
218. If any force generates motion, a double force will generate
double motion, and so on, whether simultaneously or successively,
instantaneously or gradually, applied. And this motion, if the body
was moving beforehand, is either added to the previous motion if
directly conspiring with it; or is subtracted if directly opposed ; or
is geometrically compounded with it, according to the kinematical
principles already explained, if the line of previous motion and the
direction of the force are inclined to each other at any angle. (This
is a paraphrase of Newton's own comments on the second law.)
219. In Chapter I. we have considered change of velocity, or
acceleration, as a purely geometrical element, and have seen how it
may be at once inferred from the given initial and final velocities of a
body. By the definition of a quantity of motion (§ 211), we see that,
if we multiply the change of velocity, thus geometrically determined,
by the mass of the body, we have the change of motion referred to in
Newton's law as the measure of the force which produces it.
It is to be particularly noticed, that in this statement there is nothing
said about the actual motion of the body before it was acted on by the
force : it is only the cha7ige of motion that concerns us. Thus the
same force will produce precisely the same change of motion in a
body, whether the body be at rest, or in motion with any velocity
whatever.
220. Again, it is to be noticed that nothing is said as to the body
being under the action of one force only ; so that we may logically
put a part of the second law in the following (apparently) amplified
form : —
Wheji any forces whatever act 07i a body, the7t, whether the body be
07'igi7ially at rest or moving with a7iy velocity a7id i7i a7iy direction, each
force produces in the body the exact change of 7notio7i which itivouldhave
produced if it had acted singly on the body originally at rest.
221. A remarkable consequence follows immediately from this view
of the second law. Since forces are measured by the changes of
72 . PRELIMINARY.
motion they produce, and their directions assigned by the directions
in which these changes are produced; and since the changes of
motion of one and the same body are in the directions of, and pro-
portional to, the changes of velocity — a single force, measured by the
resultant change of velocity, and in its direction, will be the equivalent
of any number of simultaneously acting forces. Hence
The resultajit of any number of forces {applied at one point) is to he
found by the same geometrical process as the resultant of any jtumber of
simultaneous velocities,
222. From this follows at once (§ 31) the construction of the
Parallelogram of Forces for finding the resultant of two forces, and
the Polygon of Forces for the resultant of any number of forces, in
lines all through one point.
The case of the equilibrium of a number of forces acting at one
point, is evidently deducible at once from this ; for if we introduce
one other force equal and opposite to their resultant, this will produce
a change of motion equal and opposite to the resultant change of
motion produced by the given forces ; that is to say, will produce a
condition in which the point experiences no change of motion, which,
as we have already seen, is the only kind of rest of which we can ever
be conscious.
223. Though Newton perceived that the Parallelogram of Forces,
or the fundamental principle of Statics, is essentially involved in the
second law of motion, and gave a proof which is virtually the same as
the preceding, subsequent writers on Statics (especially in this country)
have very generally ignored the fact ; and the consequence has been
the introduction of various unnecessary Dynamical Axioms, more or
less obvious, but in reality included in or dependent upon Newton's
laws of motion. We have retained Newton's method, not only on
account of its admirable simplicity, but because we believe it contains
the most philosophical foundation for the static as well as for the
kinetic branch of the dynamic science. ,
224. But the second law gives us the means of measuring force,
and also of measuring the mass of a body.
For, if we consider the actions of various forces upon the same
body for equal times, we evidently have changes of velocity produced
which 2XQ proportional to the forces. The changes of velocity, then,
give us in this case the means of comparing the magnitudes of different
forces. Thus the velocities acquired in one second by the same mass
(falling freely) at different parts of the earth's surface, give us the
relative amounts of the earth's attraction at these places.
Again, if equal forces be exerted on different bodies, the changes
of velocity produced in equal times must be inversely as the masses
of the various bodies. This is approximately the case, for instance,
with trains of various lengths started by the same locomotive : it is
exactly realized in such cases as the action of an electrified body on
a number of solid or hollow spheres of the same external diameter,
and of different metals.
DYNAMICAL LAWS AND PRINCIPLES. 73
Again, if we find a case in which different bodies, each acted on
by a force, acquire in the same time the same changes of velocity,
the forces must be proportional to the masses of the bodies. This,
when the resistance of the air is removed, is the case of falling bodies;
and from it we conclude that the weight of a body in any given
locality, or the force with which the earth attracts it, is proportional
to its mass ; a most important physical truth, which will be treated
of more carefully in the chapter devoted to Properties of Matter.
225. It appears, lastly, from this law, that every theorem of Kine-
matics connected with acceleration has its counterpart in Kinetics.
Thus, for instance (§ 38), we see that the force under which a par-
ticle describes any curve, may be resolved into two components, one
in the tangent to the curve, the other towards the centre of curvature ;
their magnitudes being the acceleration of momentum, and the pro-
duct of the momentum and the angular velocity about the centre of
curvature, respectively. In the case of uniform motion, the first of
these vanishes, or the whole force is perpendicular to the direction
of motion. When there is no force perpendicular to the direction
of motion, there is no curvature, or the path is a straight line.
226. We have, by means of the first two laws, arrived at a defijiitiott
and a measure of force ; and have also found how to compound, and
therefore also how to resolve, forces : and also how to investigate
the motion of a single particle subjected to given forces. But more
is required before we can completely understand the more complex
cases of motion, especially those in which we have mutual actions
between or amongst two or more bodies; such as, for instance,
attractions, or pressures, or transferrence of energy in any form.
This is perfectly supplied by
227. Lex III. Actioni contrariam semper et aeqiialem esse reactio-
nem: sive corporiim duorum actiones in se mutuo semper esse aequales
et in partes contrarias dirigi.
To every action there is always an equal and contrary reaction: or, the
mutual actio7is of any two bodies are always equal a?td oppositely directed.
228. If one body presses or draws another, it is pressed or
drawn by this other with an equal force in the opposite direction.
If any one presses a stone with his finger, his finger is pressed with
the same force in the opposite direction by the stone. A horse
towing a boat on a canal is dragged backwards by a force equal to
that which he impresses on the towing-rope forwards. By whatever
amount, and in whatever direction, one body has its motion changed
by impact upon another, this other body has its motion changed by
the same amount in the opposite direction; for at each instant during
the impact the force between them was equal and opposite on the
two. When neither of the two bodies has any rotation, whether
before or after impact, the changes of velocity which they experience
are inversely as their masses.
When one body attracts another from a distance, this other attracts
it with an equal and opposite force. This law holds not only for
74 PRELIMINARY.
the attraction of gravitation, but also, as Newton himself remarked
and verified by experiment, for magnetic attractions : also for electric
forces, as tested by Otto-Guericke.
229. What precedes is founded upon Newton's own comments
on the third law, and the actions and reactions contemplated are
simple forces. In the scholium appended, he makes the following
remarkable statement, introducing another specification of actions
and reactions subject to his third law, the full meaning of which
seems to have escaped the notice of commentators : —
Si aestitnetur agentis actio ex ejus vi et velocitate conjundim; et
similiter resistentis reactio aestimetur conjunctim ex ejus partium singu-
lariim velocitatibus et viribiis resistendi ab earwn attritione, cohaesioiie^
pondere^ et acceleratioiie oriiindis ; erunt actio et reactio^ in omni instru-
mentorimi usn, sibi invicem semper aeqiiales.
In a previous discussion Newton has shown what is to be under-
stood by the velocity of a force or resistance ; i. e. that it is the
velocity of the point of application of the force resolved in the direction
of the force, in fact proportional to the virtual velocity. Bearing this
in mind, we may read the above statement as follows : —
If the action of an agent be measured by the product of its force into
its velocity; and if similarly, the reaction of the resistaiice be 7neasured
by the velocities of its several parts into their several forces, whether
these arise from friction, cohesion, weight, or acceleration; — action and
reaction, in all combinations of machines, will be equal and opposite.
To avoid confusion it is perhaps better to use the word Activity as
the equivalent of Actio in this second specification.
Farther on we shall give a full development of the consequences
of this most important remark.
230. Newton, in the passage just quoted, points out that forces
of resistance against acceleration are to be reckoned as reactions
equal and opposite to the actions by which the acceleration is pro-
duced. Thus, if we consider any one material point of a system,
its reaction against acceleration must be equal and opposite to the
resultant of the forces which that point experiences, whether by the
actions of other parts of the system upon it, or by the influence of
matter not belonging to the system. In other words, it must be in
equilibrium with these forces. Hence Newton's view amounts to this,
that all the forces of the system, with the reactions against accelera-
tion of the material points composing it, form groups of equilibrating
systems for these points considered individually. Hence, by the
principle of superposition of forces in equilibrium, all the forces
acting on points of the system form, with the reactions against acce-
leration, an equilibrating set of forces on the whole system. This
is the celebrated principle first explicitly stated, and very usefully
applied, by D'Alembert in 1742, and still known by his name. We
have seen, however, that it is very distinctly implied in Newton's
own interpretation of his third law of motion. As it is usual to inves-
DYNAMICAL LAWS AND PRINCIPLES. 75
tigate the general equations or conditions of equilibrium, in treatises
on Analytical Dynamics, before entering in detail on the kinetic
branch of the subject, this principle is found practically most useful
in showing how we may write down at once the equations of motion
for any system for which the equations of equilibrium have been
investigated.
231. Every rigid body may be imagined to be divided into inde-
finitely small parts. Now, in whatever form we may eventually
find a physical explanation of the origin of the forces which act
between these parts, it is certain that each such small part may be
considered to be held in its position relatively to the others by mutual
forces in lines joining them.
232. From this we have, as immediate consequences of the second
and third laws, and of the preceding theorems relating to centre of
inertia and moment of momentum, a number of important propo-
sitions such as the following : —
ia) The centre of inertia of a rigid body moving in any manner,
but free from external forces, moves uniformly in a straight line.
{I)) When any forces whatever act on the body, the motion of the
centre of inertia is the same as it would have been had these forces
been applied with their proper magnitudes and directions at that
point itself.
{c) Since the moment of a force acting on a particle is the same
as the moment of momentum it produces in unit of time, the changes
of moment of momentum in any two parts of a rigid body due to
their mutual action are equal and opposite. Hence the moment of
momentum of a rigid body, about any axis which is fixed in direction,
and passes through a point which is either fixed in space or moves
uniformly in a straight line, is unaltered by the mutual actions of the
parts of the body.
id) The rate of increase of moment of momentum, when the body
is acted on by external forces, is the sum of the moments of these
forces about the axis.
233. We shall for the present take for granted, that the mutual
action between two rigid bodies may in every case be imagined as
composed of pairs of equal and opposite forces in straight lines.
From this it follows that the sum of the quantities of motion, parallel
to any fixed direction, of two rigid bodies influencing one another
in any possible way, remains unchanged by their mutual action;
also that the sum of the moments of momentum of all the particles
of the two bodies, round any line in a fixed direction in space, and
passing through any point moving uniformly in a straight line in any
direction, remains constant. From the first of these propositions we
infer that the centre of inertia of any number of mutually influencing,
bodies, if in motion, continues moving uniformly in a straight line,,
unless in so far as the direction or velocity of its motion is changed
by forces acting mutually between them and some other matter not
belonging to them ; also that the centre of inertia of any body or
76 PRELIMINARY.
system of bodies moves just as all their matter, if concentrated in
a point, would move under the influence of forces equal and parallel
to the forces really acting on its different parts. From the second
we infer that the axis of resultant rotation through the centre of
inertia of any system of bodies, or through any point either at rest
or moving uniformly in a straight line, remains unchanged in direc-
tion, and the sum of moments of momenta round it remains constant
if the system experiences no force from without. This principle
is sometimes called Cofiservation of Areas, a not very convenient
designation. From this principle it follows that if by internal action
such as geological upheavals or subsidences, or pressure of the winds
on the water, or by evaporation and rain- or snow-fall, or by any in-
fluence not depending on the attraction of 'Sun or moon (even though
dependent on solar heat), the disposition of land and water becomes
altered, the component round any fixed axis of the moment of mo-
mentum of the earth's rotation remains constant.
234. The kinetic energy of any system is equal to the sum of the
kinetic energies of a mass equal to the sum of the masses of the
system, moving with a velocity equal to that of its centre of inertia,
and of the motions of the separate parts relatively to the centre of
inertia.
Let 6>/ represent the velocity of the centre of inertia, IP that of
p any point of the system relative to O. Then
^J^ the actual velocity of that point is OP, and the
^^^^/ \ proof of § 196 applies at once — it being re-
^-^ / \ membered that the mean of 7(2, i. e. the mean
-^i— r^ i of the velocities relative to th'e centre of inertia
^ i « g^^^ parallel to 01, is zero by § 65.
235. The kinetic energy of rotation of a rigid system about any
axis is (§§ 55, 179) expressed by \%mi^<i?, where ;// is the mass of
any part, r its distance from the axis, and w the angular velocity of
rotation. It may evidently be written in the form \i^^%mt^. The
factor ^niT^ is of course (§ 198) the Moment of Inertia of the system
about the axis in question.
It is worth while to notice that the moment of momentum of any
rigid system about an axis, being '^mvr=(a%7nr^, is the product of
the angular velocity into the moment of inertia; while, as above, the
half product of the moment of inertia by the square of the angular
velocity is the kinetic energy.
If we take a quantity k, such that
k'^m^-^mr^
k is called the Radius of Gyration about the axis from which r is
measured. The radius of gyration about any axis is therefore the
distance from that axis at which, if the whole mass were placed, it
would have the same moment of inertia as before. In a fly-wheel,
where it is desirable to have as great a moment of inertia with as
small a mass as possible, within certain limits of dimensions, the
D YNAMICAL LA WS AND PRINCIPLES. 7 7
greater part of the mass is formed into a ring of the largest admis-
sible diameter, and the radius of this ring is then approximately the
radius of gyration of the whole.
236. The rate of increase of moment of momentum is thus, in New-
ton's notation (§ 28), wi%jnr^; and, in the case of a body free to rotate
about a fixed axis, is equal to the moment of the couple about that
axis. Hence a constant couple gives uniform acceleration of angular
velocity; or <b= . By § 178 we see that the corresponding
Force
formula for linear acceleration is s = v = — — ^ .
M
237. For every rigid body there may be described about any point
as centre, an ellipsoid (called Poinsofs Momenta! Ellipsoid) which is
such that the length of any radius-vector is inversely proportional to
the radius of gyration of the body about that radius- vector as axis.
The axes of the ellipsoid are the Principal Axes of inertia of the
body at the point in question.
When the moments of inertia about two of these are equal, the
ellipsoid becomes a spheroid, and the radius of gyration is the same
for every axis in the plane of its equator.
When all three principal moments are equal, the ellipsoid becomes
a sphere, and every axis has the same radius of gyration.
238. The principal axes at any point of a rigid body are normals
to the three surfaces of the second order which pass through that
point, and are confocal with an ellipsoid, having its centre at the
centre of inertia, and its three principal diameters coincident with the
three principal axes through these points, and equal respectively to
the doubles of the radii of gyration round them. This ellipsoid is
called the Cetitral Ellipsoid.
239. A rigid body is said to be kinetically symmetrical about its
centre of inertia when its moments of inertia about three principal
axes through that point are equal ; and therefore necessarily the
moments of inertia about all axes through that point equal (§ 237),
and all these axes principal axes. About it uniform spheres, cubes,
and in general any complete crystalline solid of the first system (see
chapter on Properties of Matter) are kinetically symmetrical.
A rigid body is kinetically symmetrical about an axis when this
axis is one of the principal axes through the centre of inertia, and
the moments of inertia about the other two, and therefore about any
line in their plane, are equal. A spheroid, a square or equilateral
triangular prism or plate, a circular ring, disc, or cylinder, or any
complete crystal of the second or fourth system, is kinetically sym-
metrical about its axis.
240. The foundation of the abstract theory of energy is laid by
Newton in an admirably distinct and compact manner in the sentence
of his scholium already quoted (§ 229), in which he points out its
78 PRELIMINARY.
application to mechanics^. The actio agentis, as he defines it, which
is evidently equivalent to the product of the effective component 'of
the force, into the velocity of the point on which it acts, is simply, in
modern English phraseology, the rate at which the agent works. The
subject for measurement here is precisely the same as that for which
Watt, a hundred years later, introduced the practical unit of a ^Horse-
p07ver^^ or the rate at which an agent works when overcoming 33,000
times the weight of a pound through the space of a foot in a minute ;
that is, producing 550 foot-pounds of work per second. The unit,
however, which is most generally convenient is that which Newton's
definition implies, namely, the rate of doing work in which the unit
of energy is produced in the unit of time.
241. Looking at Newton's words (§ 229) in this light, we see that
they may be logically converted into the following form : —
Work done o?i any system of bodies (in Newton's statement, the parts
of any machine) has its equivalent in work done against friction^
molecular forces, or gravity, if there be no accele?'ation ; but if there
be acceleration, part of the work is expended in overcoming the resistance
to acceleration, and the additional kinetic energy developed is equivalent
to the work so spent. This is evident from § 180.
When part of the work is done against molecular forces, as in
bending a spring; or against gravity, as in raising a weight; the
recoil of the spring, and the fall of the weight, are capable at any
future time, of reproducing the work originally expended (§ 207).
But in Newton's day, and long afterwards, it was supposed that work
was absolutely lost by friction; and, indeed, this statement is still to
be found even in recent authoritative treatises. But we must defer
the examination of this point till we consider in its modern form the
principle of Co?iservation of Energy.
242. If a system of bodies, given either at rest or in motion, be
influenced by no forces from without, the sum of the kinetic energies
of all its parts is augmented in any time by an amount equal to the
whole work done in that time by the mutual forces, which we may
imagine as acting between its points. When the lines in which these
forces act remain all unchanged in length, the forces do no work, and
the sum of the kinetic energies of the whole system remains constant.
If, on the other hand, one of these lines varies in length during the
motion, the mutual forces in it will do work, or will consume work,
according as the distance varies with or against them.
243. A limited system of bodies is said to be dynamically con-
servative (or simply cotiservative, when force is understood to be the
subject), if the mutual forces between its parts always perform, or
always consume, the same amount of work during any motion
1 The reader will remember that we use the word 'mechanics' in its true classical
sense, the science of machines, the sense in which Newton himself used it, when he
dismissed the further consideration of it by saying (in the scholium referred to),
Caeferum viecJianicani h-nctare non rst Jinjiis iusfifuti.
DYNAMICAL LA WS AND PRINCIPLES. 79
whatever, by which it can pass from one particular configuration
to another.
244. The whole theory of energy in physical science is founded
on the following proposition : —
If the mutual forces between the parts of a material system are
independent of their velocities, whether relative to one another, or
relative to any external matter, the system must be dynamically
conservative.
For if more work is done by the mutual forces on the different
parts of the system in passing from one particular configuration to
another, by one set of paths than by another set of paths, let the
system be directed, by frictionless constraint, to pass from the first
configuration to the second by one set of paths and return by the
other, over and over again for ever. It will be a continual source of
energy without any consumption of materials, which is impossible.
245. The potential energy of a conservative system, in the confi-
guration which it has at any instant, is the amount of work that its
mutual forces perform during the passage of the system from any
one chosen configuration to the configuration at the time referred to.
It is generally, but not always, convenient to fix the particular con-
figuration chosen for the zero of reckoning of potential energy, so
that the potential energy, in every other configuration practically
considered, shall be positive.
246. The potential energy of a conservative system, at any instant,
depends solely on its configuration at that instant, being, according to
definition, the same at all times when the system is brought again
and again to the same configuration. It is therefore, in mathematical
language, said to be a function of the co-ordinates by which the
positions of the different parts of the system are specified. If, for
example, we have a conservative system consisting of two material
points; or two rigid bodies, acting upon one another with force
dependent only on the relative position of a point belonging to one
of them, and a point belonging to the other; the potential energy
of the system depends upon the co-ordinates of one of these points
relatively to lines of reference in fixed directions through the other.
It will therefore, in general, depend on three independent co-ordi-
nates, which we may conveniently take as the distance between the
two points, and two angles specifying the absolute direction of the
line joining them. Thus, for example, let the bodies be two uniform
metal globes, electrified with any given quantities of electricity, and
placed in an insulating medium such as air, in a region of space
under the influence of a vast distant electrified body. The mutual
action between these two spheres will depend solely on the relative
position of their centres. It will consist partly of gravitation, de-
pending solely on the distance between their centres, and of electric
force, which will depend on the distance between them, but also, in
virtue of the inductive action of the distant body, will depend on the
absolute direction of the line joining their centres. Or again, if the
8o PRELIMINARY.
system consist of two balls of soft iron, in any locality of the earth's
surface, their mutual action will be partly gravitation, and partly
due to the magnetism induced in them by terrestrial magnetic force.
The portion of the potential energy depending on the latter cause,
will be a function of the distance between their centres and the in-
clination of this line to the direction of the terrestrial magnetic force,
247. In nature the hypothetical condition of § 243 is apparmtly
violated in all circumstances of motion. A material system can never
be brought through any returning cycle of motion without spending
more work against the mutual forces of its parts than is gained from
these forces, because no relative motion can take place without
meeting with frictional or other forms of resistance; among which
are included (i) mutual friction between solids sliding upon one
another; (2) resistances due to the viscosity of fluids, or imperfect
elasticity of solids; (3) resistances due to the induction of electric
currents; (4) resistances due to varying magnetization under the
influence of imperfect magnetic retentiveness. No motion in nature
can take place without meeting resistance due to some, if not to all,
of these influences. It is matter of everyday experience that friction
and imperfect elasticity of solids impede the action of all artificial
mechanisms; and that even when bodies are detached, and left to
move freely in the air, as falling bodies, or as projectiles, they expe-
rience resistance owing to the viscosity of the air.
The greater masses, planets and comets, moving in a less resisting
medium, show less indications of resistance ^ Indeed it cannot be said
that observation upon any one of these bodies, with the possible excep-
tion of Encke's comet, has demonstrated resistance. But the analogies
of nature, and the ascertained facts of physical science, forbid us to
doubt that every one of them, every star, and every body of any kind
moving in any part of space, has its relative motion impeded by the
air, gas, vapour, medium, or whatever we choose to call the substance
occupying the space immediately round it; just as the motion of a
rifle-bullet is impeded by the resistance of the air.
248. There are also indirect resistances, owing to friction impeding
the tidal motions, on all bodies which, like the earth, have portions
of their free surfaces covered by liquid, which, as long as these bodies
move relatively to neighbouring bodies, must keep drawing ofl" energy
from their relative motions. Thus, if we consider, in the first place,
the action of the moon alone, on the earth with its oceans, lakes, and
rivers, we perceive that it must tend to equalize the periods of the
earth's rotation about its axis, and of the revolution of the two bodies
about their centre of inertia ; because as long as these periods differ,
the tidal action of the earth's surface must keep subtracting energy
from their modons. To view the subject more in detail, and, at the
same time, to avoid unnecessary complications, let us suppose the
1 Newton, Principia. (Remarks on the first law of motion.) 'Majora auteni
Planetarum et Cometarum corpora motus suos et progressivos et circulares, in
spatiis minus resistentibus factos, conservant diutius.'
DYNAMICAL LA WS AND PRINCIPLES. 8i
moon to be a uniform spherical body. The mutual action and
reaction of gravitation between her mass and the earth's, will be
equivalent to a single force in some line through her centre • and
must be such as to impede the earth's rotation as long as this is
performed in a shorter period than the moon's motion round the
earth. It must therefore lie in some such direction as the line MQ
in the diagram, which represents, necessarily
with enormous exaggeration, its deviation,
OQ, from the earth's centre. Now the actual
force on the moon in the line MQ^ may be
regarded as consisting of a force in the line
MO towards the earth's centre, sensibly
equal in amount to the whole force, and a
comparatively very small force in the line
MT perpendicular to MO. This latter is
very nearly tangential to the moon's path,
and is in the direction with her motion.
Such a force, if suddenly commencing to act, would, in the first place,
increase the moon's velocity; but after a certain time she would have
moved so much farther from the earth, in virtue of this acceleration,
as to have lost, by moving against the earth's attraction, as much
velocity as she had gained by the tangential accelerating force. The
integral effect on the moon's motion, of the particular disturbing
cause now under consideration, is most easily found by using the prin-
ciple of moments of momenta (§ 233). Thus we see that as much
moment of momentum is gained in any time by the motions of the
centres of inertia of the moon and earth relatively to their common
centre of inertia, as is lost by the earth's rotation about its axis. It
is found that the distance would be increased to about 347,100 miles,
and the period lengthened to 48-36 days. Were there no other body
in the universe but the earth and the moon, these two bodies might
go on moving thus for ever, in circular orbits round their common
centre of inertia, and the earth rotating about its axis in the same
period, so as always to turn the same face to the moon, and therefore
to have all the liquids at its surface at rest relatively to the solid. But
the existence of the sun would prevent any such state of things from
being permanent. There would be solar tides — twice high water and
twice low water — in the period of the earth's revolution relatively to
the sun (that is to say, twice in the solar day, or, which would be the
same thing, the month). This could not go on without loss of energy
by fluid friction. It is not easy to trace the whole course of the
disturbance in the earth's and moon's motions which this cause
would produce, but its ultimate effect must be to bring the earth,
moon, and sun to rotate round their common centre of inertia, like
parts of one rigid body. It is probable that the moon, in ancient
times liquid or viscous in its outer layer if not throughout, was thus
brought to turn always the same face to the earth.
249. We have no data in the present state of science for estimating
the relative importance of tidal friction, and of the resistance of the
T. 6
82 PRELIMINARY.
resisting medium through which the earth and moon move; but what-
ever it may be, there can be but one ultimate result for such a system
as that of the sun and planets, if continuing long enough under ex-
isting laws, and not disturbed by meeting with other moving masses
in space. That result is the falling together of all into one mass,
which, although rotating for a time, must in the end come to rest
relatively to the surrounding medium.
250. The theory of energy cannot be completed until we are able
to examine the physical influences which accompany loss of energy
in each of the classes of resistance mentioned above (§ 247). We
shall then see that in every case in which energy is lost by resistance,
heat is generated; and we shall learn from Joule's investigations that
the quantity of heat so generated is a perfectly definite equivalent for
the energy lost. Also that in no natural action is there ever a develop-
ment of energy which cannot be accounted for by the disappearance
of an equal amount elsewhere by means of some known physical
agency. Thus we shall conclude, that if any Hmited portion of the
material universe could be perfectly isolated, so as to be prevented
from either giving energy to, or taking energy from, matter external
to it, the sum of its potential and kinetic energies would be the same
at all times: in other words, that every material system subject to no
other forces than actions and reactions between its parts, is a dyna-
mically conservative system, as defined above (§ 243). But it is only
when the inscrutably minute motions among small parts, possibly the
ultimate molecules of matter, which constitute light, heat, and mag-
netism; and the intermolecular forces of chemical affinity; are taken
into account, along with the palpable motions and measurable forces
of which we become cognizant by direct observation, that we can
recognize the universally conservative character of all natural dynamic
action, and perceive the bearing of the principle of reversibility on the
whole class of natural actions involving resistance, which seem to
violate it. In the meantime, in our studies of abstract dynamics, it
will be sufficient to introduce a special reckoning for energy lost in
working against, or gained from work done by, forces not belonging
"palpably to the conservative class.
251. The only actions and reactions between the parts of a system,
not belonging palpably to the conservative class, which we shall con-
sider in abstract dynamics, are those of friction between soHds sliding
on solids, except in a few instances in which we shall consider the
general character and ultimate results of effects produced by viscosity
of fluids, imperfect elasticity of solids, imperfect electric conduction,
or imperfect magnetic retentiveness. We shall also, in abstract dyna-
mics, consider forces as applied to parts of a limited system arbitrarily
from without. These we shall call, for brevity, the applied forces.
252. The law of energy may then, in abstract dynamics, be ex-
pressed as follows : —
The whole work done in any time, on any limited material system,
by applied forces, is equal to the whole effect in the forms of potential
DYNAMICAL LAWS AND PRINCIPLES. 83
and kinetic energy produced in the system, together with the work lost
in friction.
253. This principle may be regarded as comprehending the whole
of abstract dynamics, because, as we now proceed to show, the con-
ditions of equihbrium and of motion, in every possible case, may be
derived from it.
254. A material system, whose relative motions are unresisted by
friction, is in equilibrium in any particular configuration if, and is not
in equilibrium unless, the rate at which the applied forces perform
work at the instant of passing through it is equal to that at which
potential energy is gained, in every possible motion through that
configuration. This is the celebrated principle of virtual velocities
which Lagrange made the basis of his Mecaniqiie Analytiqiie.
255. To prove it, we have first to remark that the system cannot
possibly move away from any particular configuration except by work
being done upon it by the forces to which it is subject: it is therefore
in equilibrium if the stated condition is fulfilled. To ascertain that
nothing less than this condition can secure the equilibrium, let us
first consider a system having only one degree of freedom to move.
Whatever forces act on the whole system, we may always hold it in
equilibrium by a single force applied to any one point of the system
in its line of motion, opposite to the direction in which it tends to
move, and of such magnitude that, in any infinitely small motion in
either direction, it shall resist, or shall do, as much work as the other
forces, whether applied or internal, altogether do or resist. Now, by
the principle of superposition of forces in equilibrium, we might,
without altering their effect, apply to any one point of the system such
a force as we have just seen would hold the system in equilibrium, and
another force equal and opposite to it. All the other forces being
balanced by one of these two, they and it might again, by the principle
of superposition of forces in equilibrium, be removed; and therefore
the whole set of given forces would produce the same effect, whether
for equilibrium or for motion, as the single force which is left acting
alone. This single force, since it is in a line in which the point of its
application is free to move, must move the system. Hence the given
forces, to which the single force has been proved equivalent, cannot
possibly be in equilibrium unless their whole work for an infinitely
small motion is nothing, in which case the single equivalent force is
reduced to nothing. But whatever amount of freedom to move the
whole system may have, we may always, by the application of fric-
tionless constraint, limit it to one degree of freedom only; — and this
may be freedom to execute any particular motion whatever, possible
under the given conditions of the system. If, therefore, in any such
infinitely small motion, there is variation of potential energy uncom-
pensated by work of the applied forces, constraint limiting the freedom
of the system to only this motion will bring us to the case in which we
have just demonstrated there cannot be equilibrium. But the applica-
6—2
84 PRELIMINARY.
tion of constraints limiting motion cannot possibly disturb equilibrium,
and therefore the given system under the actual conditions cannot be
in equilibrium in any particular configuration if the rate of doing work
is greater than that at which potential energy is stored up in any pos-
sible motion through that configuration.
256. If a material system, under the influence of internal and
applied forces, varying according to some definite law, is balanced
by them in any position in which it may be placed, its equilibrium is
said to be neutral. This is the case with any spherical body of
uniform material resting on a horizontal plane. A right cylinder or
cone, bounded by plane ends perpendicular to the axis, is also in
neutral equilibrium on a horizontal plane. Practically, any mass of
moderate dimensions is in neutral equilibrium when its centre of
inertia only is fixed, since, when its longest dimension is small in
comparison with the earth's radius, gravity is, as we shall see, ap-
proximately equivalent to a single force through this point.
But if, when displaced infinitely little in any direction from a par-
ticular position of equilibrium, and left to itself, it commences and
continues vibrating, without ever experiencing more than infinitely
small deviation in any of its parts, from the position of equilibrium,
the equilibrium in this position is said to be stable. A weight sus-
pended by a string, a uniform sphere in a hollow bowl, a loaded sphere
resting on a horizontal plane with the loaded side lowest, an oblate
body resting with one end of its shortest diameter on a horizontal
plane, a plank, whose thickness is small compared with its length and
breadth, floating on water, are all cases of stable equilibrium; if we
neglect the motions of rotation about a vertical axis in the second,
third, and fourth cases, and horizontal motion in general, in the fifth,
for all of which the equilibrium is neutral.
If, on the other hand, the system can be displaced in any way from
a position of equilibrium, so that when left to itself it will not vibrate
within infinitely small limits about the position of equilibrium, but will
move farther and farther away from it, the equilibrium in this position
is said to be unstable. Thus a loaded sphere resting on a horizontal
plane with its load as high as possible, an egg-shaped body standing
on one end, a board floating edgewise in water, would present, if they
could be realized in practice, cases of unstable equilibrium.
When, as in many cases, the nature of the equilibrium varies with
the direction of displacement, if unstable for any possible displace-
ment it is practically unstable on the whole. Thus a circular disc
standing on its edge, though in neutral equiHbrium for displacements
in its plane, yet being in unstable equilibrium for those perpendicular
to its plane, is practically unstable. A sphere resting in equilibrium on
a saddle presents a case in which there is stable, neutral, or unstable
equilibrium, according to the direction in which it may be displaced
by rolling; but practically it is unstable.
257. The theory of energy shows a very clear and simple test for
discriminating these characters, or determining whether the equilibrium
DYNAMICAL LAWS AND PRINCIPLES. 85
is neutral, stable, or unstable, in any case. If there is just as much
potential energy stored up as there is work performed by the applied and
internal forces in any possible displacement, the equilibrium is neutral,
but not unless. If in every possible infinitely small displacement
from a position of equihbrium there is more potential energy stored
up than work done, the equilibrium is thoroughly stable, and not
unless. If in any or in every infinitely small displacement from a
position of equilibrium there is more work done than energy stored
up, the equilibrium is unstable. It follows that if the system is in-
fluenced only by internal forces, or if the applied forces follow the
law of doing always the same amount of work upon the system pass-
ing from one configuration to another by all possible paths, the whole
potential energy must be constant, in all positions, for neutral equili-
brium; must be a minimum for positions of thoroughly stable equili-
brium; must be either a maximum for all displacements, or a maximum
for some displacements and a minimum for others, when there is
unstable equilibrium.
258. We have seen that, according to D'Alembert's principle, as
explained above (§ 230), forces acting on the different points of a
material system, and their reactions against the accelerations which
they actually experience in any case of motion, are in equiUbrium
with one another. Hence in any actual case of motion, not only is
the actual work done by the forces equal to the kinetic energy pro-
duced in any infinitely small time, in virtue of the actual accelerations;
but so also is the work which would be done by the forces, in any
infinitely small time, if the velocities of the points constituting the
system were at any instant changed to any possible infinitely small
velocities, and the accelerations unchanged. This statement, when
put into the concise language of mathematical analysis, constitutes
Lagrange's application of the * principle of virtual velocities' to ex-
press the conditions of D'Alembert's equilibrium between the forces
acting, and the resistances of the masses to acceleration. It com-
prehends, as we have seen, every possible condition of every case of
motion. The 'equations of motion' in any partix:ular case are, as
Lagrange has shown, deduced from it with great ease.
259. When two bodies, in relative motion, come into contact,
pressure begins to act between them to prevent any parts of them
from jointly occupying the same space. This force commences from
nothing at the first point of collision, and gradually increases per unit
of area on a gradually increasing surface of contact. If, as is always
the case in nature, each body possesses some degree of elasticity, and
if they are not kept together after the impact by cohesion, or by some
artificial appliance, the mutual pressure between them will reach a
maximum, will begin to diminish, and in the end will come to nothing,
by gradually diminishing in amount per unit of area on a gradually
diminishing surface of contact. The whole process would occupy
not greatly more or less than an hour if the bodies were of such
dimensions as the earth, and such degrees of rigidity as copper, steel,
86 PRELIMINARY.
or glass. It is finished, probably, within a thousandth of a second,
if they are globes of any of these substances not exceeding a yard
in diameter.
260. The whole amount, and the direction, of the * Impact' Qx^t-
rienced by either body in any such case, are reckoned according to
the * change of momentum' which it experiences. The amount of
the impact is measured by the amount, and its direction by the
direction of the change of momentum, which is produced. The
component of an impact in a direction parallel to any fixed line is
similarly reckoned according to the component change of momentum
in that direction.
261. If we imagine the whole time of an impact divided into
a very great number of equal intervals, each so short that the force
does not vary sensibly during it, the component change of momentum
in any direction during any one of these intervals will (§ 185) be
equal to the force multiplied by the measure of the interval. Hence
the component of the impact is equal to the sum of the forces in all
the intervals, multiplied by the length of each interval.
262. Any force in a constant direction acting in any circumstances,
for any time great or small, may be reckoned on the same principle ;
so that what we may call its whole amount during any time, or its
* time-integral^^ will measure, or be measured by, the whole momentum
which it generates in the time in question. But this reckoning is not
often convenient or useful except when the whole operation con-
sidered is over before the position of the body, or configuration of
the system of bodies, involved, has altered to such a degree as to
bring any other forces into play, or alter forces previously acting,
to such an extent as to produce any sensible effect on the momentum
measured. Thus if a person presses gently with his hand, during
a few seconds, upon a mass suspended by a cord or chain, he pro-
duces an effect which, if we know the degree of the force at each
instant, may be thoroughly calculated on elementary principles. No
approximation to a full determination of the motion, or to answering
such a partial question as ' how great will be the whole deflection
produced?' can be founded on a knowledge of the ^time-integral*
alone. If, for instance, the force be at first very great and gradually
diminish, the effect will be very different from what it would be if the
force were to increase very gradually and to cease suddenly, even
although the time-integral were the same in the two cases. But if
the same body is ' struck a blow,' in a horizontal direction, either by
the hand, or by a mallet or other somewhat hard mass, the action
of the force is finished before the suspending cord has experienced
any sensible deflection from the vertical. Neither gravity nor any
other force sensibly alters the effect of the blow. And therefore the
whole momentum at the end of the blow is sensibly equal to the
* amount of the impact,' which is, in this case, simply the time-
integral.
DYNAMICAL LAWS AND PRINCIPLES. 87
263. Such Is the case of Robins' Ballistic Pendulum^ a massive
block of wood movable about a horizontal axis at a considerable
distance above it — employed to measure the velocity of a cannon or
musket-shot. The shot is fired into the block in a horizontal direc-
tion perpendicular to the axis. The impulsive penetration is so
nearly instantaneous, and the inertia of the block so large compared
with the momentum of the shot, that the ball and pendulum are
moving on as one mass before the pendulum has been sensibly deflected
from the position of equilibrium. This is the essential peculiarity of the
ballistic method ; which is used also extensively in electro-magnetic
researches and in practical electric testing, when the integral quantity
of the electricity which has passed in a current of short duration is to
be measured. The ballistic formula (§ 272) is appHcable, with the
proper change of notation, to all such cases.
264. Other illustrations of the cases in which the time-integral
gives us the complete solution of the problem may be given without
limit. They include all cases in which the direction of the force is
always coincident with the direction of motion of the moving body,
and those special cases in which the time of action of the force is so
short that the body's motion does not, during its lapse, sensibly alter
its relation to the direction of the force, or the action of any other
forces to which it may be subject. Thus, in the vertical fall of a
body, the time-integral gives us at once the change of momentum ;
and the same rule applies in most cases of forces of brief duration,
as in a * drive ' in cricket or golf.
265. The simplest case which we can consider, and the one usually
treated as an introduction to the subject, is that of the collision of
two smooth spherical bodies whose centres before colHsion were
moving in the same straight line. The force between them at each
instant must be in this line, because of the symmetry of circumstances
round it ; and by the third law it must be equal in amount on the
two bodies. Hence (Lex II.) they must experience changes of
motion at equal rates in contrary directions ; and at any instant of
the impact the integral amounts of these changes of motion must be
equal. Let us suppose, to fix the ideas, the two bodies to be moving
both before and after impact in the same direction in one line : one
of them gaining on the other before impact, and either following it
at a less speed, or moving along with it, as the case may be, after
the impact is completed. Cases in which the former is driven back-
wards by the force of the collision, or in which the two moving in
opposite directions meet in collision, are easily reduced to dependence
on the same formula by the ordinary algebraic convention with regard
to positive and negative signs.
In the standard case, then, the quantity of motion lost, up to any
instant of the impact, by one of the bodies, is equal to that gained
by the other. Hence at the instant when their velocities are equalized
they move as one mass with a momentum equal to the sum of the
88 PRELIMINARY.
momenta of the two before Impact. That is to say, if v denote the
common velocity at this instant, we have
{M-\-M')v=:MV-^M'V\
MV+M'V
or
M+M'
if M, M' denote the masses of the two bodies, and K, V their
velocities before impact.
During this first period of the impact the bodies have been, on
the whole, coming into closer contact with one another, through a
compression or deformation experienced by each, and resulting, as
remarked above, in a fitting together of the two surfaces over a
finite area. No body in nature is perfectly inelastic ; and hence,
at the instant of closest approximation, the mutual force called
into action between the two bodies continues, and tends to separate
them. Unless prevented by natural surface cohesion or welding (such
as is always found, as we shall see later in our chapter on Properties
of Matter, however hard and well polished the surfaces may be), or
by artificial appliances (such as a coating of wax, applied in one of
the common illustrative experiments; or the coupling applied between
two railway-carriages when run together so as to push in the springs,
according to the usual practice at railway-stations), the two bodies are
actually separated by this force, and move away from one another.
Newton found t\i2it, provided the impact is not so violent as to make any
sensible permanent indentation in either body, the relative velocity of
separation after the impact bears a proportion to their previous
relative velocity of approach, which is constant for the same two
bodies. This proportion, always less than unity, approaches more
and more nearly to it the harder the bodies are. Thus with balls of
compressed wool he found it f, iron nearly the same, glass if. The
results of more recent experiments on the same subject have con-
firmed Newton's law. These will be described later. In any case
of the collision of two balls, let e denote this proportion, to which we
give the name Coefficient of Restitution^ ; and, with previous nota-
tion, let in addition U, U' denote the velocities of the two bodies
after the conclusion of the impact ; in the standard case each being
positive, but U' > U. Then we have
U'- u=e{y- V'\
and, as before, since one has lost as much momentum as the other
has gained, mU^ M' U' = MV^ M' V,
From these equations we find
(J/-f M')U= MF+ M'r-eM'{V- V),
with a similar expression for U'.
1 In most modern treatises this is called a 'coefficient of elasticity;' a
misnomer, suggested, it may be, by Newton's words, but utterly at variance with
modern language and modern knowledge regarding elasticity.
DYNAMICAL LAWS AND PRINCIPLES. 89
Also we have, as above,
Hence, by subtraction,
(J/+J/')(z/- U)=eM'{y- V')=e{M'V-{M^M')v + MV\,
and therefore ^ _ ^= e ( V- v).
Of course we have also U' -v = e{v- V).
These results may be put in words thus : — The relative velocity of
either of the bodies with regard to the centre of inertia of the two
is, after the completion of the impact, reversed in direction, and
diminished in the ratio e -. \.
266. Hence the loss of kinetic energy, being, according to §§ 233,
234, due only to change of kinetic energy relative to the centre of
inertia, is to this part of the whole as 1 - e^ \ i.
Thus by § 234,
Initial kinetic energy = | {M -^ M')v' + 1 J/ ( F- v)' + ^M' (v - F')\
Final „ „ =i{M+M')v^ + ^M{v-U'y + ^M'(C/'-vy.
Loss = ^ (i - ^) { Jf ( F-vf + JkI{v- Vy}.
267. When two elastic bodies, the two balls supposed above for
instance, impinge, some portion of their previous kinetic energy will
always remain in them as vibrations. A portion of the loss of energy
(miscalled the effect of imperfect elasticity alone) is necessarily due
to this cause in every real case.
Later, in our chapter on the Properties of Matter, it will be shown
as a result of experiment, that forces of elasticity are, to a very close
degree of accuracy, simply proportional to the strains (§ 135), within
the limits of elasticity, in elastic solids which, like metals, glass, etc.,
bear but small deformations without permanent change. Hence when
two such bodies come into collision, sometimes with greater and
sometimes with less mutual velocity, but with all other circumstances
similar, the velocities of all particles of either body, at corresponding
times of the impacts, will be always in the same proportion. Hence
the velocity of separation of the centres of inertia after impact will
bear a constant proportion to the previous velocity of approach ;
which agrees with the Newtonian law. It is therefore probable that
a very sensible portion, if not the whole, of the loss of energy in the
visible motions of two elastic bodies, after impact, experimented on
by Newton, may have been due to vibrations ; but unless some other
cause also was largely operative, it is difficult to see how the loss was
so much greater with iron balls than with glass.
268. In certain definite extreme cases, imaginable although not
realizable, no energy will be spent in vibrations, and the two bodies
will separate, each moving simply as a rigid body, and having in this
simple motion the whole energy of work done on it by elastic force
during the collision. For instance, let the two bodies be cylinders,
or prismatic bars with flat ends, of the same kind -of substance, and of
96 . PRELIMINARY,
equal and similar transverse sections ; and let this substance have the
property of compressibility with perfect elasticity, in the direction of
the length of the bar, and of absolute resistance to change in every
transverse dimension. Before impact, let the two bodies be placed
with their lengths in one line, and their transverse sections (if not
circular) similarly situated, and let one or both be set in motion in
this line. Then, if the lengths of the two be equal, they will separate
after impact with the same relative velocity as that with which they
approached, and neither will retain any vibratory motion after the
end of the collision. The result, as regards the motions of the two
bodies after the collision, will be sensibly the same if they are of any
real ordinary elastic solid material, provided the greatest transverse
diameter of each is very small in comparison of its length.
269. If the two bars are of an unequal length, the shorter will, after
the impact, be in exactly the same state as if it had struck another
of its own length, and it therefore will move as a rigid body after the
collision. But the other will, along with a motion of its centre of
gravity, calculable from the principle that its whole momentum must
(§ '^Z'i) be changed by an amount equal exactly to the momentum
gained or lost by the first, have also a vibratory motion, of which the
whole kinetic and potential energy will make up the deficiency of
energy which we shall presently calculate in the motions of the centres
of inertia. For simplicity, let the longer body be supposed to be at
rest before the collision. Then the shorter on striking it will be left
at rest ; this being clearly the result in the case of the ^ = i in the
preceding formulae (§ 265) applied to the impact of one body striking
another of equal mass previously at rest. The longer bar will move
away with the same momentum, and therefore with less velocity of its
centre of inertia, and less kinetic energy of this motion, than the other
body had before impact, in the ratio of the smaller to the greater
mass. It will also have a very remarkable vibratory motion, which,
when its length is more than double of that of the other, will consist
of a wave running backwards and forwards through its length, and
causing the motion of its ends, and, in fact, of every particle of it, to
take place by ' fits and starts,' not continuously. The full analysis of
these circumstances, though very simple, must be reserved until we
are especially occupied with waves, and the kinetics of elastic solids.
It is sufficient at present to remark, that the motions of the centres of
inertia of the two bodies after impact, whatever they may have been
previously, are given by the preceding formulae with for e the value
M'
-^ , where M and M' are the smaller and larger mass respectively.
270. The mathematical theory of the vibrations of solid elastic
spheres has not yet been worked out ; and its application to the case
of the vibrations produced by impact presents considerable difficulty.
Experiment, however, renders it certain, that but a small part of the
whole kinetic energy of the previous motions can remain in the form
of vibrations after the impact of two equal spheres of glass or of
DYNAMICAL LAWS AND PRINCIPLES, 91
ivory. This is proved, for instance, by the common observation, that
one of them remains nearly motionless after striking the other pre-
viously at rest; since, the velocity of the common centre of inertia of
the two being necessarily unchanged by the impact, we infer that the
second ball acquires a velocity nearly equal to that which the first had
before striking it. But it is to be expected that unequal balls of the
same substance coming into collision will, by impact, convert a very
sensible proportion of the kinetic energy of their previous motions
into energy of vibrations ; and generally, that the same will be the
case when equal or unequal masses of different substances come into
collision ; although for one particular proportion of their diameters,
depending on their densities and elastic qualities, this effect will be
a minimum, and possibly not much more sensible than it is when the
substances are the same and the diameters equal.
271. It need scarcely be said that in such cases of impact as that
of the tongue of a bell, or of a clock-hammer striking its bell (or
spiral spring as in the American clocks), or of pianoforte-hammers
striking the strings, or of the drum struck with the proper implement,
a large part of the kinetic energy of the blow is spent in generating
vibrations.
272. The Moment of aft Impact about any axis is derived from the
line and amount of the impact in the same way as the moment of
a velocity or force is determined from the line and amount of the
velocity or force, § 46. If a body is struck, the change of its
moment of momentum about any axis is equal to the moment of the
impact round that axis. But, without considering the measure of the
impact, we see (§ 233) that the moment of momentum round any axis,
lost by one body in striking another, is, as in every case of mutual
action, equal to that gained by the other.
Thus, to recur to the ballistic pendulum — the line of motion of the
bullet at impact may be in any direction whatever, but the only part
which is effective is the component in a plane perpendicular to the
axis. We may therefore, for simplicity, consider the motion to be in
a line perpendicular to the axis, though not necessarily horizontal.
Let m be the mass of the bullet, v its velocity, and / the distance of
its line of motion from the axis. Let M be the mass of the pendulum
with the bullet lodged in it, and k its radius of gyration. Then if <o
be the angular velocity of the pendulum when the impact is complete,
mvp = Mk^dij
from which the solution of the question is easily determined.
For the kinetic energy after impact is changed (§ 207) into its
equivalent in potential energy when the pendulum reaches its position
of greatest deflection. Let this be given by the angle 6 : then the
height to which the centre of inertia is raised is/i(i - cos 6) if A be its
distance from the axis. Thus
MgA (t- cos 6) ^-iMiW^i"^^,
^""Jw
9 2 PRELIMINAR Y,
. 6 m p V m p TTV .^ -,
or 2Sin-= 7>.Y.— 7== -r>.^. -^, IX 7^
2 M k Jg/i M h gT'
an expression for the chord of the angle of deflection. In practice
the chord of the angle 6 is measured by means of a light tape or
cord attached to a point of the pendulum, and slipping with small
friction through a clip fixed close to the position occupied by that
point when the pendulum hangs at rest.
273. Work do7ie by an impact is, in general, the product of the
impact into half the sum of the initial and final velocities of the point
at which it is applied, resolved in the direction of the impact. In the
case of direct impact, such as that treated in § 265, the initial kinetic
energy of the body is ^MF^ the final ^MZ/^, and therefore the gain
by the impact is
or, which is the same.
But M{U- V) is (§ 260) equal to the amount of the impact. Hence
the proposition : the extension of which to the most general cir-
cumstances is not difficult, but requires somewhat higher analysis
than can be admitted here.
274. It is worthy of remark, that if any number of impacts be
applied to a body, their whole effect will be the same whether they
be applied together or successively (provided that the whole time
occupied by them be infinitely short), although the work done by
each particular impact is, in general, different according to the order
in which the several impacts are applied. The whole amount of
work is the sum of the products obtained by multiplying each impact
by half the sum of the components of the initial and final velocities
of the point to which it is applied.
275. The effect of any stated impulses, applied to a rigid body,
or to a system of material points or rigid bodies connected in any
way, is to be found most readily by the aid of D'Alembert's principle;
according to which the given impulses, and the impulsive reaction
against the generation of motion, measured in amount by the
momenta generated, are in equilibrium; and are, therefore, to be
dealt with mathematically by applying to them the equations of
equilibrium of the system.
276. [A material system of any kind, given at rest, and subjected
to an impulse in any specified direction, and of any given magnitude,
moves off so as to take the greatest amount of kinetic energy which
the specified impulse can give it.
277. If the system is guided to take, under the action of a given
impulse, any motion different from the natural motion, it will have
less kinetic energy than that of the natural motion, by a difference
equal to the kinetic energy of the motion represented by the resultant
(§ 67) of those two motions, one of them reversed.
DYNAMICAL LA WS AND PRINCIPLES. 93
Cor. If a set of material points are struck independently by
impulses each given in amount, more kinetic energy is generated if
the points are perfectly free to move each independently of all the
others, than if they are connected in any way. And the deficiency
of energy in the latter case is equal to the amount of the kinetic
energy of the motion which geometrically compounded with the
motion of either case would give that of the other.
278. Given any material system at rest. Let any parts of it be
set in motion suddenly with any specified velocities, possible accord-
ing to the conditions of the system; and let its other parts be
influenced only by its connexions with those. It is required to
find the motion. The solution of the problem is — The motion
actually taken by the system is that which has less kinetic energy than
any other motion fulfilling the prescribed velocity conditions. And
the excess of the energy of any other such motion, above that of the
actual motion, is equal to the energy of the motion that would be
generated by the action alone of the impulse which, if compounded
with the impulse producing the actual motion, would produce this
other supposed motion.]
279. Maupertuis' celebrated principle of Least Action has been,
even up to the present time, regarded rather as a curious and some-
what perplexing property of motion, than as a useful guide in kinetic
investigations. We are strongly impressed with the conviction that
a much more profound importance will be attached to it, not only
in abstract dynamics, but in the theory of the several branches of
physical science now beginning to receive dynamic explanations.
As an extension of it. Sir W. R. Hamilton' has evolved his method
of Varying Action^ which undoubtedly must become a most valuable
aid in future generalizations.
What is meant by ' Action' in these expressions is, unfortunately,
something very different from the Actio Agent is defined by Newton,
and, it must be admitted, is a much less judiciously chosen word.
Taking it, however, as we find it now universally used by writers on
dynamics, we define the Action of a Moving System as proportional
to the average kinetic energy, which it has possessed during the time
from any convenient epoch of reckoning, multiplied by the time.
According to the unit generally adopted, the action of a system
which has not varied in its kinetic energy, is twice the amount of the
energy multiplied by the time from the epoch. Or if the energy has
been sometimes greater and sometimes less, the action at time /
is the double of what we may call the time-integral of the energy;
that is to say, the action of a system is equal to the sum of the
average momenta for the spaces described by the particles from any
era each multiplied by the length of its path.
280. The principle of Least Action is this : — Of all the different
sets of paths along which a conservative system may be guided to
move from one configuration to another, with the sum of its potential
1 Phil. Trans., 1834— 1835.
94 PRELIMINARY.
and kinetic energies equal to a given constant, that one for which the
action is the least is such that the system will require only to be
started with the proper velocities, to move along it unguided.
281. [In any unguided motion whatever, of a conservative system,
the Action from any one stated position to any other, though not
necessarily a minimum, fulfils the stationary condition, that is to say,
the condition that the variation vanishes, which secures either a
minimum or maximum, or maximum-minimum.]
282. From this principle of stationary action, founded, as we have
seen, on a comparison between a natural motion, and any other
motion, arbitrarily guided and subject only to the law of energy, the
initial and final configurations of the system being the same in each
case; Hamilton passes to the consideration of the variation of the
action in a natural or unguided motion of the system produced by
varying the initial and final configurations, and the sum of the
potential and kinetic energies. The result is, that —
283. The rate of decrease of the action per unit of increase of
any one of the free (generalized) co-ordinates specifying the
initial configuration, is equal to the corresponding (generalized) com-
ponent momentum of the actual motion from that configuration :
the rate of increase of the action per unit increase of any one
of the free co-ordinates specifying the final configuration, is equal
to the corresponding component momentum of the actual motion
towards this second configuration : and the rate of increase of the
action per unit increase of the constant sum of the potential and
kinetic energies, is equal to the time occupied by the motion of
which the action is reckoned.
284. The determination of the motion of any conservative system
from one to another of any two configurations, when the sum of its
potential and kinetic energies is given, depends on the determination
of a single function of the co-ordinates specifying those configura-
tions by means of two quadratic, partial differential equations of the
first order, with reference to those two sets of co-ordinates respec-
tively, with the condition that the corresponding terms of the two
differential equations become separately equal when the values of
the two sets of co-ordinates agree. The function thus determined
and employed to express the solution of the kinetic problem was
called the Characteristic Function, by Sir W. R. Hamilton, to whom
the method is due. It is, as we have seen, the 'action' from one
of the configurations to the other; but its peculiarity in Hamilton's
system is, that it is to be expressed as a function of the co-ordinates
and a constant, the whole energy, as explained above. It is evidently
symmetrical witli respect to the two configurations, changing only in
sign if their co-ordinates are interchanged.
■ 285. The most general possible solution of the quadratic, partial
differential equation of the first order, satisfied by Hamilton's Cha-
DYNAMICAL LA WS AND PRINCIPLES, 95
racteristic Function (either terminal configuration alone varying),
when interpreted for the case of a single free particle, expresses the
action up to any point from some point of a certain arbitrarily
given surface, from which the particle has been projected, in the
direction of the normal, and with the proper velocity to make the
sum of the potential and actual energies have a given value. In other
words, the physical problem solved by the most general solution of
that partial differential equation, for a single free particle, is this : —
Let free particles, not mutually influencing one another, be pro-
jected normally from all points of a certain arbitrarily given surface,
each with the proper velocity to make the sum of its potential and
kinetic energies have a given value. To find, for that one of the
particles which passes through a given point, the 'action' in its course
from the surface of projection to this point. The Hamiltonian
principles stated above, show that the surfaces of equal action cut
the paths of the particles at right angles; and give also the
following remarkable properties of the motion : —
If, from all points of an arbitrary surface, particles not mutually
influencing one another be projected with the proper velocities in
the directions of the normals; points which they reach with equal
actions lie on a surface cutting the paths at right angles. The
infinitely small thickness of the space between any two such surfaces
corresponding to amounts of action differing by any infinitely small
quantity, is inversely proportional to the velocity of the particle
traversing it ; being equal to the infinitely small diflerence of action
divided by the whole momentum of the particle.
286. Irrespectively of methods for finding the 'characteristic
function' in kinetic problems, the fact that any case of motion what-
ever can be represented by means of a single function in the manner
explained in § 284, is most remarkable, and, when geometrically
interpreted, leads to highly important and interesting properties of
motion, which have valuable applications in various branches of
Natural Philosophy ; one of which, explained below, led Hamilton *
to a general theory of optical instruments, comprehending the whole
in one expression. Some of the most direct applications of the
general principle to the motions of planets, comets, etc., considered
as free points, and to the celebrated problem of perturbations, known
as the Problem of Three Bodies, are worked out in considerable detail
by Hamilton {Phil. Trans. ^ 1834-5), and in various memoirs by
Jacobi, Liouville, Bour, Donkin, Cayley, Boole, etc.
The now abandoned, but still interesting, corpuscular theory of
light furnishes the most convenient language for expressing the
optical application. In this theory light is supposed to consist of
material particles not mutually influencing one another ; but subject
to molecular forces from the particles of bodies, not sensible at
sensible distances, and therefore not causing any deviation from
uniform rectilinear motion in a homogeneous medium, except within
1 On the Theory of Systems of Rays. Trans. R. I. A., 1824, 1830, 1832.
96 PRELIMINARY.
an indefinitely small distance from its boundary. The laws of reflec-
tion and of single refraction follow correctly from this hypothesis,
which therefore suffices for what is called geometrical optics.
We hope to return to this subject, with sufficient detail, in treating
of Optics. At present we limit ourselves to state a theorem com-
prehending the known rule for measuring the magnifying power of
a telescope or microscope (by comparing the diameter of the object-
glass with the diameter of pencil of parallel rays emerging from the
eye-piece, when a point of light is placed at a great distance in front
of the object-glass), as a particular case.
287. Let any number of attracting or repelling masses, or perfectly
smooth elastic objects, be fixed in space. Let two stations, O and O',
be chosen. Let a shot be fired with a stated velocity, V, from (7,
in such a direction as to pass through O'. There may clearly be
more than one natural path by which this maybe done; but, generally
speaking, when one such path is chosen, no other, not sensibly di-
verging from it, can be found; and any infinitely small deviation in
the line of fire from (9, will cause the bullet to pass infinitely near to,
but not through O'. Now let a circle, with infinitely small radius r,
be described round O as centre, in a plane perpendicular to the line
of fire from this point, and let — all with infinitely nearly the same
velocity, but fulfilling the condition that the sum of the potential and
kinetic energies is the same as that of the shot from O — bullets be
fired from all points of this circle, all directed infinitely nearly parallel
to the line of fire from O, but each precisely so as to pass through O'.
Let a target be held at an infinitely small distance, «', beyond Cf,
in a plane perpendicular to the line of the shot reaching it from O.
The bullets fired from the circumference of the circle round (9, will,
after passing through 0\ strike this target in the circumference of an
exceedingly small ellipse, each with a velocity (corresponding of
course to its position, under the law of energy) difiering infinitely
little from V\ the common velocity with which they pass through O'.
Let now a circle, equal to the former, be described round 0\ in the
plane perpendicular to the central path through 0\ and let bullets be
fired from points in its circumference, each with the proper velocity,
and in such a direction infinitely nearly parallel to the central path
as to make it pass through O. These bullets, if a target is held to
y
receive them perpendicularly at a distance a = a' -^,, beyond O, will
strike it along the circumference of an ellipse equal to the former
and placed in a corresponding position; and the points struck by the
individual bullets will correspond in the manner explained below.
Let F and F' be points of the first and second circles, and Q and
Q the points on the first and second targets which bullets from them
strike ; then if F' be in a plane containing the central path through
(y, and the position which Q would take if its ellipse were made
circular by a pure strain (§ 159) ; Q and Q are similarly situated on
the two ellipses.
DYNAMICAL LAWS AND PRINCIPLES. 97
288. The most obvious optical application of this remarkable
result is, that in the use of any optical apparatus whatever, if the eye
and the object be interchanged without altering the position of the
instrument, the magnifying power is unaltered. This is easily under-
stood when, as in an ordinary telescope, microscope, or opera-glass
(Galilean telescope), the instrument is symmetrical about an axis, and
is curiously contradictory of the common idea that a telescope 'dimi-
nishes' when looked through the wrong way, which no doubt is true
if the telescope is simply reversed about the middle of its length, eye
and object remaining fixed. But if the telescope be removed from the
eye till its eye-piece is close to the object, the part of the object seen
will be seen enlarged to the same extent as when viewed with the
telescope held in the usual manner. This is easily verified by looking
from a distance of a few yards, in through the object-glass of an
opera-glass, at the eye of another person holding it to his eye in
the usual way.
The more general application may be illustrated thus: — Let the
points, (9, O' (the centres of the two circles described in the preceding
enunciation), be the optic centres of the eyes of two persons looking
at one another through any set of lenses, prisms, or transparent
media arranged in any way between them. If their pupils are of
equal sizes in reality, they will be seen as similar ellipses of equal
apparent dimensions by the two observers. Here the imagined
particles of light, projected from the circumference of the pupil of
either eye, are substituted for the projectiles from the circumference
of either circle, and the retina of the other eye takes the place of the
target receiving them, in the general kinetic statement.
289. If instead of one free particle we have a conservative system
of any number of mutually influencing free particles, the same state-
ment may be applied with reference to the initial position of one of
the particles and the final position of another, or with reference to the
initial positions, or to the final positions, of two of the particles. It
thus serves to show how the influence of an infinitely small change in
one of those positions, on the direction of the other particle passing
through the other position, is related to the influence on the direction
of the former particle passing through the former position produced
by an infinitely small change in the latter position, and is of immense
use in physical astronomy. A corresponding statement, in terms of
generalized co-ordinates, may of course be adapted to a system of
rigid bodies or particles connected in any way. All such statements
are included in the following very general proposition: —
The rate of increase of the component momentum relative to
any one of the co-ordinates, per unit of increase of any other co-
ordinate, is equal to the rate of increase of the component momentum
relative to the latter per unit increase or diminution of the former
co-ordinate, according as the two co-ordinates chosen belong to one
configuration of the system, or one of them belongs to the initial
configuration and the other to the final.
r. 7
98 PRELIMINARY.
290. If a conservative system is infinitely little displaced from a
configuration of stable equilibrium, it will ever after vibrate about this
configuration, remaining infinitely near it; each particle of the system
performing a motion which is composed of simple harmonic vibra-
tions. If there are i degrees of freedom to move, and we consider
any system of generalized co-ordinates specifying its position at
any time, the deviation of any one of these co-ordinates from its
value for the configuration of equilibrium will vary according to a
complex harmonic function (§ 88), composed in general of i simple
harmonics of incommensurable periods, and therefore (§ 85) the whole
motion of the system will not recur periodically through the same
series of configurations. There are in general, however, i distinct
determinate displacements, which we shall call the normal displace-
ments, fulfilling the condition, that if any one of them be produced
alone, and the system then left to itself for an instant at rest, this
displacement will diminish and increase periodically according to
a simple harmonic function of the time, and consequently every
particle of the system will execute a simple harmonic movement in
the same period. This result, we shall see later, includes cases in
which there are an infinite number of degrees of freedom; as, for
instance, a stretched cord; a mass of air in a closed vessel; waves
in water, or oscillations of water in a vessel of limited extent, or an
elastic solid; and in these applications it gives the theory of the
so-called 'fundamental vibration,' and successive 'harmonics' of the
cord, and of all the different possible simple modes of vibration in
the other cases. In all these cases it is convenient to give the name
* fundamental mode' to any one of the possible simple harmonic
vibrations, and not to restrict it to the gravest simple harmonic mode,
as has been hitherto usual in respect to vibrating cords and organ-
pipes.
The whole kinetic energy of any complex motion of the system is
equal to the sum of the kinetic energies of the fundamental constitu-
ents; and the potential energy of any displacements is equal to the sum
of the potential energies of its normal components. Corresponding
theorems of normal constituents and fundamental modes of motion,
and the summation of their kinetic and potential energies in complex
motions and displacements, hold for motion in the neighbourhood of
a configuration of U7istable equilibrium. In this case, some or all of
the constituent motions are fallings away from the position of equi-
librium (according as the potential energies of the constituent normal
vibrations are negative).
291. If, as may be in particular cases, the periods of the vibrations
for two or more of the normal displacements are equal, any displace-
ment compounded of them will also fulfil the condition of a normal
displacement. And if the system be displaced according to any one
such normal displacement, and projected with velocity corresponding
to another, it will execute a movement, the resultant of two simple
harmonic movements in equal periods. The graphic representation
DYNAMICAL LA WS AND PRINCIPLES. 99
of the variation of the corresponding co-ordinates of the system, laid
down as two rectangular co-ordinates in a plane diagram, will con-
sequently (§ 82) be a circle or an ellipse; which will therefore, of
course, be the form of the orbit of any particle of the system which
has a distinct direction of motion, for two of the displacements in
question. But it must be remembered that some of the principal
parts may have only one degree of freedom ; or even that each part
of the system may have only one degree of freedom (as, for instance,
if the system is composed of a set of particles each constrained to
remain on a given line, or of rigid bodies on fixed axes, mutually
influencing one another by elastic cords or otherwise). In such a
case as the last, no particle of the system can move otherwise than
in one line; and the ellipse, circle, or other graphical representation
of the composition of the harmonic motions of the system, is merely
an aid to comprehension, and not a representation of any motion
actually taking place in any part of the system.
292. In nature, as has been said above (§ 250), every system
uninfluenced by matter external to it is conservative, when the
ultimate molecular motions constituting heat, light, and magnetism,
and the potential energy of chemical affinities, are taken into account
along with the palpable motions and measurable forces. But (§ 247)
practically we are obliged to admit forces of friction, and resistances
of the other classes there enumerated, as causing losses of energy to
be reckoned, in abstract dynamics, without regard to the equivalents
of heat or other molecular actions which they generate. Hence when
such resistances are to be taken into account, forces opposed to the
motions of various parts of a system must be introduced into the
equations. According to the approximate knowledge which we have
from experiment, these forces are independent of the velocities when
due to the friction of sofids; and are simply proportional to the
velocities when due to fluid viscosity directly, or to electric or magnetic
influences, with corrections depending on varying temperature, and
on the varying configuration of the system. In consequence of the
last-mentioned cause, the resistance of a real liquid (which is always
more or less viscous) against a body moving very rapidly through it,
and leaving a great deal of irregular motion, such as 'eddies,' in its
wake, seems to be nearly in proportion to the square of the velocity;
although, as Stokes has shown, at the lowest speeds the resistance
is probably in simple proportion to the velocity, and for all speeds
may, it is probable, be approximately expressed as the sum of two
terms, one simply as the velocity, and the other as the square of the
velocity. If a solid is started from rest in an incompressible fluid,
the initial law of resistance is no doubt simple proportionality to the
velocity, (however great, if suddenly enough given;) until by the
gradual growth of eddies the resistance is increased gradually till it
comes to fulfil Stokes's law.
293. The effect of friction of solids rubbing one against another
is simply to render impossible the infinitely small vibrations with which
7—2
loo PRELIMINARY,
we are now particularly concerned ; and to allow any system in which
it is present, to rest balanced when displaced within certain finite
limits, from a configuration of frictionless equilibrium. In mechanics
it is easy to estimate its effects with sufficient accuracy when any
practical case of finite oscillations is in question. But the other
classes of dissipative agencies give rise to resistances simply as the
velocities, without the corrections referred to, when the motions are
infinitely small, and can never balance the system in a configuration
deviating to any extent, however small, from a configuration of
equilibrium without friction. In the theory of infinitely small vibra-
tions, they are to be taken into account by adding to the expressions
for the generalized components of force, terms consisting of the
generalized velocities each multiplied by a constant, which gives us
equations still remarkably amenable to rigorous mathematical treat-
ment. The result of the integration for the case of a single degree
of freedom is very simple; and it is of extreme importance, both for
the explanation of many natural phenomena, and for use in a large
variety of experimental investigations in Natural Philosophy. Partial
conclusions from it, in the first place, stated in general terms, are
as follows: —
294. If the resistance per unit velocity is less than a certain limit,
in any particular case, the motion is a- simple harmonic oscillation,
with amplitude decreasing by equal proportions in equal successive
intervals of time. But if the resistance exceeds this limit, the system,
when displaced from its position of equilibrium and left to itself,
returns gradually towards its position of equilibrium, never oscillating
through it to the other side, and only reaching it after an infinite
time.
In the unresisted motion, let «' be the rate of acceleration, when
the displacement is unity; so that (§ 74) we have T=^ — ; and let the
rate of retardation due to the resistance corresponding to unit velocity
be k. Then the motion is of the oscillatory or non-oscillatory class
according as k^ <{2nf or k'^-> {2tif. In the first case, the period of
n
the oscillation is increased, by the resistance, from 7" to Tj~^ — TT^Yi*
and the rate at which the Napierian logarithm of the amplitude
diminishes per unit of time is \k.
295. An indirect but very simple proof of this important propo-
sition may be obtained by means of elementary mathematics as
follows : — A point describes a logarithmic spiral with uniform angular
velocity about the pole — find the acceleration.
Since the angular velocity of SP and the inclination of this line
to the tangent are each constant, the linear velocity of P is as SP.
Take a length PT^ equal to n SP, to represent it. Then the
hodograph, the locus of /, where »S/ is parallel and equal to PT, is
evidently another logarithmic spiral similar to the former, and de-
scribed with the same uniform angular velocity. Hence (§§ 35, 49)
DYNAMICAL LA WS AND PRINCIPLES. loi
pt, the acceleration required, is equal to n Sp, and makes with Sp an
angle Spt equal to SPT. Hence, if Pu be drawn parallel and equal
to //, and uv parallel to PT, the whole acceleration // or Pu may be
resolved into Pv and vu. Now Pvu is an isosceles triangle, whose
base angles {v, u) are each equal to the constant angle of the spiral.
Hence Pv and vu bear constant ratios to Puj and therefore to SP
and /'T' respectively.
The acceleration, therefore, is composed of a central attractive
part proportional to the distance, and a tangential retarding part
proportional to the velocity.
And, if the resolved part of /*'s motion parallel to any line in the
plane of the spiral be considered, it is obvious that in it also the
acceleration will consist of two parts — one directed towards a point
in the line (the projection of the pole of the spiral), and proportional
to the distance from it ; the other proportional to the velocity, but
retarding the motion.
Hence a particle which, unresisted, would have a simple harmonic
motion, has, when subject to resistance proportional to its velocity,
a motion represented by the resolved part of the spiral motion just
described.
296. If a be the constant angle of the spiral, w the angular velocity
of SP, we have evidently
PT, sin a = SP.o), But PT= nSP, so that n = -A- .
sma
Hence Pv = Pu =// = nSp = nPT= n' . SP
and vu = 2Pv . cos a = 2« cos aPT= k . PT (suppose).
2
Thus the central force at unit distance is n^= . ^ , and the co-
sm a
cc. • I. r ' ^ • r 2(0 cos a
emcient of resistance \s k= 211 cos a = — -. .
sma
27r
The time of oscillation in the resolved motion is evidently -- ; but,
(U
if there had been no resistance, the properties of simple harmonic
I03 PRELIMINARY.
motion show that it would have been — : so that it is increased by
I ^'
the resistance in the ratio cosec a to i , or f^ to k n^ — .
V 4
The rate of diminution of SP is evidently
PT, C0Sa = ;2 cos a SP^-SP',
2
that is, SP diminishes in geometrical progression as time increases,
k
the rate being - per unit of time per unit of length. By an ordinary
result of arithmetic (compound interest payable every instant) the
k
diminution of log . SP in unit of time is - .
° 2
This process of solution is only applicable to resisted harmonic
vibrations when n is greater than - . When ;/ is not greater than -
the auxiliary curve can no longer be a logarithmic spiral, for the
moving particle never describes more than a finite angle about the
pole. A curve, derived from an equilateral hyperbola, by a process
somewhat resembling that by which the logarithmic spiral is deduced
from a circle, may be introduced ; but then the geometrical method
ceases to be simpler than the analytical one, so that it is useless to
pursue the investigation farther, at least from this point of view.
297. The general solution of the problem, to find the motion of
a system having any number, /, of degrees of freedom, when infinitely
little disturbed from a position of equilibrium, and left to move subject
to resistances proportional to velocities, shows that the whole motion
may be resolved, in general determinately, into 2/ different motions
each either simple harmonic with amplitude diminishing according to
the law stated above (§ 294), or non-oscillatory, and consisting of
equi-proportionate diminutions of the components of displacement
in equal successive intervals of time.
298. When the forces of a system depending on configuration,
and not on motion, or, as we may call them for brevity, the forces
of position, violate the law of conservatism, we have seen (§ 244)
that energy without limit may be drawn from it by guiding it per-
petually through a returning cycle of configurations, and we have
inferred that in every real system, not supplied with energy from
without, the forces of position fulfil the conservative law. But it is
easy to arrange a system artificially, in connexion with a source of
energy, so that its forces of position shall be non-conservative ; and
the consideration of the kinetic effects of such an arrangement,
especially of its oscillations about or motions round a configuration
of equilibrium, is most instructive, by the contrasts which it presents
to the phenomena of a natural system.
299. But although, when the equilibrium is stable, no possible
DYNAMICAL LA WS AND PRINCIPLES. 103
infinitely small displacement and velocity given to the system can
cause it, when left to itself, to go on moving either farther and farther
away till a finite displacement is reached, or till a finite velocity is
acquired; it is very remarkable that stability should be possible,
considering that even in the case of stability an endless increase of
velocity may, as is easily seen from § 244, be obtained merely by
constraining the system to a particular closed course, or circuit of
configurations, nowhere deviating by more than an infinitely small
amount from the configuration of equilibrium, and leaving it at
rest anywhere in a certain part of this circuit. This result, and
the distinct peculiarities of the cases of stability and instability, are
sufticiently illustrated by the simplest possible example, — that of a
material particle moving in a plane.
300. There is scarcely any question in dynamics more important
for Natural Philosophy than the stability or instability of motion. We
therefore, before concluding this chapter, propose to give some
general explanations and leading principles regarding it.
A 'conservative disturbance of motion' is a disturbance in the
motion or configuration of a conservative system, not altering the
sum of the potential and kinetic energies. A conservative disturb-
ance of the motion through any particular configuration is a change
in velocities, or component velocities, not altering the whole kinetic
energy. Thus, for example, a conservative disturbance of the motion
of a particle through any point, is a change in the direction of its
motion, unaccompanied by change of speed.
301. The actual motion of a system, from any particular con-
figuration, is said to be stable if every possible infinitely small con-
servative disturbance of its motion through that configuration may
be compounded of conservative disturbances, any one of which would
give rise to an alteration of motion which would bring the system
again to some configuration belonging to the undisturbed path, in
a finite time, and without more than an infinitely small digression.
If this condition is not fulfilled, the motion is said to be unstable.
302. For example, if a body. A, be supported on a fixed vertical
axis ; if a second, B^ be supported on a parallel axis belonging to
the first; a third, C, similarly supported on B^ and so on; and if
B, C, etc., be so placed as to have each its centre of inertia as far as
possible from the fixed axis, and the whole set in motion with
a common angular velocity about this axis, the motion will be
thoroughly stable. If, for instance, each of the bodies is a flat
rectangular board hinged on one edge, it is obvious that the whole
system will be kept stable by centrifugal force, when all are in one
plane and as far out from the axis as possible. But if A consists
partly of a shaft and crank, as a common spinning-wheel, or the fly-
wheel and crank of a steam-engine, and if B be supported on the
crank-pin as axis, and turned inwards (towards the fixed axis, or
across the fixed axis), then, even although the centres of inertia of C,
I04 PRELIMINARY.
Z>, etc., are placed as far from the fixed axis as possible, consistent
with this position of B^ the motion of the system will be unstable.
303. The rectilinear motion of an elongated body lengthwise, or
of a flat disc edgewise, through a fluid is unstable. But the motion of
either body, with its length or its broadside perpendicular to the
direction of motion, is stable. Observation proves the assertion we
have just made, for real fluids, air and water, and for a great variety
of circumstances affecting the motion; and we shall return to the
subject later, as being not only of great practical importance, but
profoundly interesting, and by no means difficult in theory.
304. The motion of a single particle affords simpler and not less
instructive illustrations of stability and instability. Thus if a weight,
hung from a fixed point by a light inextensible cord, be set in motion
so as to describe a circle about a veitical line through its position of
equilibrium, its motion is stable. For, as we shall see later, if dis-
turbed infinitely little in direction without gain or loss of energy, it
will describe a sinuous path, cutting the undisturbed circle at points
successively distant from one another by definite fractions of the
circumference, depending upon the angle of inclination of the string
to the vertical. When this angle is very small, the motion is sensibly
the same as that of a particle confined to one plane moving under
the influence of an attractive force towards a fixed point, simply pro-
portional to the distance ; and the disturbed path cuts the undisturbed
circle four times in a revolution. Or if a particle confined to one
plane, move under the influence of a centre in this plane, attracting
with a force inversely as the square of the distance, a path infinitely
little disturbed from a circle will cut the circle twice in a revolution.
Or if the law of central force be the nxkv power of the distance, and if
fz + 3 be positive, the disturbed path will cut the undisturbed circular
TT
orbit at successive angular intervals, each equal to . . But the
V// + 3
motion will be unstable if n be negative, and - « > 3.
305. The case of a particle moving on a smooth fixed surface
under the influence of no other force than that of the constraint, and
therefore always moving along a geodetic line of the surface, affords
extremely simple illustrations of stability and instability. For instance,
a particle placed on the inner circle of the surface of an anchor-ring,
and projected in the plane of the ring, would move perpetually in that
circle, but unstably, as the smallest disturbance would clearly send it
away from this path, never to return until after a digression round the
outer edge. (We suppose of course that the particle is held to the
surface, as if it were placed in the infinitely narrow space between a
solid ring and a hollow one enclosing it.) But if a particle is placed
on the outermost, or greatest, circle of the ring, and projected in its
plane, an infinitely small disturbance will cause it to describe a sinuous
path cutting the circle at points round it successively distant by angles
DYNAMICAL LAWS AND PRINCIPLES. 105
each equal to tt / - , and therefore at intervals of time, each equal to
- / - , where a denotes the radius of that circle, <o the angular velocity
m^ a
in it, and b the radius of the circular cross section of the ring. This
is proved by remarking that an infinitely narrow band from the outer-
most part of the ring has, at each point, a and b from its principal
radii of curvature, and therefore (§ 134) has for its geodetic lines the
great circles of a sphere of radius ^ab, upon which it may be bent.
306. In all these cases the undisturbed motion has been circular
or rectilineal, and, when the motion has been stable, the effect of a
disturbance has been pef-iodic, or recurring with the same phases in
equal successive intervals of time. An illustration of thoroughly stable
motion in which the effect of a disturbance is not * periodic,' is pre-
sented by a particle sliding down an inclined groove under the action
of gravity. To take the simplest case, we may consider a particle
sliding down along the lowest straight line of an inclined hollow
cylinder. If slightly disturbed from this straight line, it will oscillate
on each side of it perpetually in its descent, but not with a uniform
periodic motion, though the durations of its excursions to each side of
the straight line are all equal.
307. A very curious case of stable motion is presented by a particle
constrained to remain on the surface of an anchor-ring fixed in a
vertical plane, and projected along the great circle from any point of
it, with any velocity. An infinitely small disturbance will give rise to
a disturbed motion of which the path will cut the vertical circle over
and over again for ever, at unequal intervals of time, and unequal
angles of the circle ; and obviously not recurring periodically in any
cycle, except with definite particular values for the whole energy, some
of which are less and an infinite number are greater than that which
just suffices to bring the particle to the highest point of the ring. The
full mathematical investigation of these circumstances would afibrd an
excellent exercise in the theory of differential equations, but it is not
necessary for our present illustrations.
308. In this case, as in all of stable motion with only two degrees
of freedom, which we have just considered, there has been stability
throughout the motion ; and an infinitely small disturbance from any
point of the motion has given a disturbed path which intersects the
undisturbed path over and over again at finite intervals of time.
But, for the sake of simpHcity, at present confining our attention to
two degrees of freedom, we have a Ihnited stability in the motion of an
unresisted projectile, which satisfies the criterion of stability only at
points of its upward, not of its downward, path. Thus if MOPQ be
the path of a projectile, and if at O it be disturbed by an infinitely
small force either way perpendicular to its instantaneous direction of
motion, the disturbed path will cut the undisturbed infinitely near
the point /'where the direction of motion is perpendicular to that at 0\
ic6 PRELIMINARY.
as we easily see by considering that the line joining two particles pro-
jected from one point at the same instant with equal velocities in the
directions of any two lines, will always remain perpendicular to the
line bisecting the angle between these two.
309. The principle of varying action gives a mathematical criterion
for stability or instability in every case of motion. Thus in the first
place it is obvious {§§ 308, 311), that if the action is a true minimum
in the motion of a system from any one configuration to the con-
figuration reached at any other time, however much later, the motion
is thoroughly unstable. For instance, in the motion of a particle con-
strained to remain on a smooth fixed surface, and uninfluenced by
gravity, the action is simply the length of the path, multiplied by the
constant velocity. Hence in the particular case of a particle unin-
fluenced by gravity, moving round the inner circle in the plane of an
anchor-ring considered above, the action, or length of path, is clearly
a minimum for any one point to the point reached at any subsequent
time. (The action is not merely a minimum, but is the least possible,
from any point of the circular path to any other, through less than half
a circumference of the circle.) On the other hand, although the path
from any point in the greatest circle of the ring to any other at a dis-
tance from it along the circle, less than tt ^ab^ is clearly least possible
if along the circumference ; the path of absolutely least length is not
along the circumference between two points at a greater circular
distance than irjab from one another, nor is the path along the
circumference between them a minimum at all in this latter case. On
any surface whatever which is everywhere anticlastic, or along a geo-
detic of any surface which passes altogether through an anticlastic
region, the motion is thoroughly unstable. For if it were stable from
any point O, we should have the given undisturbed path, and the
disturbed path from O cutting it at some point Q — two difterent
DYNAMICAL LAWS AND PRINCIPLES. 107
geodetic lines joining two points ; which is impossible on any
anticlastic surface, inasmuch as the sum of the exterior angles of
any closed figure of geodetic lines exceeds four right angles when
the integral curvature of the enclosed area is negative, which
is the case for every portion of surface thoroughly anticlastic.
But, on the other hand, it is easily proved that if we have an endless
rigid band of curved surface everywhere synclastic, with a geodetic
line running through its middle, the motion of a particle projected
along this line will be stable throughout, and an infinitely slight disturb-
ance will give a disturbed path cutting the given undisturbed path again
and again for ever at successive distances differing according to the
different specific curvatures of the intermediate portions of the surface.
310. If, from any one configuration, two courses differing infinitely
little from one another, have again a configuration in common, this
second configuration will be called a kinetic focus relatively to the
first: or (because of the reversibility of the motion) these two con-
figurations will be called conjugate kinetic foci. Optic foci, if for
a moment we adopt the corpuscular theory of light, are included aa
a particular case of kinetic foci in general. But it is not difficult
to prove that there must be finite intervals of space and time be-
tween two conjugate foci in every motion of every kind of system,
only provided the kinetic energy does not vanish.
311. Now it is obvious that, provided only a sufficiently short
course is considered, the action, in any natural motion of a system,
is less than for any other course between its terminal configurations.
It will be proved presently (§ 314) that the first configuration up to
which the action, reckoned from a given initial configuration, ceases
to be a minimum, is the first kinetic focus; and conversely, that when
the first kinetic focus is passed, the action, reckoned from the initial
configuration, ceases to be a minimum ; and therefore of course can
never again be a minimum, because a course of shorter action,
deviating infinitely little from it, can be found for a part, without
altering the remainder of the whole, natural course.
312. In such statements as this it will frequently be convenient
to indicate particular configurations of the system by single letters,
as (9, P, Q, P ; and any particular course, in which it moves through
configurations thus indicated, will be called the course O...P...Q...P.
The action in any natural course will be denoted simply by the
terminal letters, taken in the order of the motion. Thus OP will
denote the action from O to P ; and therefore OP = -PO. When
there are more real natural courses from O to P than one, the
analytical expression for OP will have more than one real value;
and it may be necessary to specify for which of these courses the
action is reckoned. Thus we may have
OP for O...P...P,
OP for O...P\..P,
OP for O...P"...P,
three different values of one algebraic irrational expression.
io8
PRELIMINARY.
313. In terms of this notation the preceding statement (§311)
may be expressed thus : — If, for a conservative system, moving on
a certain course 0...F...0' ...P\ the first kinetic focus conjugate
to (9 be 0\ the action OP, in this course, will be less than the action
along any other course deviating infinitely little from it: but, on the
other hand, OP' is greater than the actions in some courses from
O to P' deviating infinitely little from the specified natural course
O...P...O'...P'.
314. It must not be supposed that the action along OP is neces-
sarily t/je least possible from O to P. There are, in fact, cases in
which the action ceases to be least of all possible, before a kinetic
focus is reached. Thus if OEAPO'E'A' be a sinuous geodetic line
cutting the outer circle of an anchor-ring, or the equator of an oblate
spheroid, in successive points O, A, A' ,\\. is easily seen that 0\ the
first kinetic focus conjugate to O, must lie somewhat beyond A.
But the length OEAP, although a minimum (a stable position for
a stretched string), is not the shortest distance on the surface from
O to jP, as this must obviously be a line lying entirely on one side
of the great circle. From O to any point, Q, short of A^ the distance
along the geodetic OEQA is clearly the least possible : but if Q be
near enough to A (that is to say, between A and the point in which
the envelop of the geodetics drawn from O, cuts OEA), there will
also be two other geodetics from O to Q. The length of one of
these will be a minimum, and that of the other not a minimum.
If Q is moved forward to A, the former becomes OE^A, equal and
similar to OEA, but on the other side of the great circle : and the
latter becomes the great circle from O to A. If now Q be moved
on to P, beyond A, the minimum geodetic OEAP ceases to be the
less of the two minima, and the geodetic OEPly'mg altogether on the
other side of the great circle becomes the least possible line from
O to P. But until Pis advanced beyond the point O', in which it
is cut by another geodetic from O lying infinitely nearly along it,
the length OEAP remains a minimum according to the general
proposition of § 311.
DYNAMICAL LAWS AND PRINCIPLES. 109
315. As it has been proved that the action from any configuration
ceases to be a minimum at the first conjugate kinetic focus, we see
immediately that if O' be the first kinetic focus conjugate to (9, reached
after passing O, no two configurations on this course from O to O'
can be kinetic foci to one another. For, the action from O just
ceasing to be a minimum when O' is reached, the action between any
two intermediate configurations of the same course is necessarily a
minimum.
316. When there are i degrees of freedom to move there are in
general, on any natural course from any particular configuration, (7,
at least /- i kinetic foci conjugate to O. Thus, for example, on the
course of a ray of light emanating from a luminous point (9, and pass-
ing through the centre of a convex lens held obliquely to its path,
there are two kinetic foci conjugate to (7, as defined above, being the
points in which the line of the central ray is cut by the so-called
*focal lines'^ of a pencil of rays diverging from O and made con-
vergent after passing through the lens. But some or all of these
kinetic foci may be on the course previous to O ', as, for instance, in
the case of a common projectile when its course passes obliquely
downwards through O. Or some or all may be lost, as when,
in the optical illustration just referred to, the lens is only strong
enough to produce convergence in one of the principal planes, or
too weak to produce convergence in either. Thus also in the
case of the undisturbed rectilineal motion of a point, or in the
motion of a point uninfluenced by force, on an anticlastic surface
(§ 309)? there are no real kinetic foci. In the motion of a pro-
jectile (not confined to one vertical plane) there can be only one
kinetic focus on each path, conjugate to one given point ; though
there are three degrees of freedom. Again, there may be any number
more than i—\ of foci in one course, all conjugate to one con-
figuration, as for instance on the course of a particle, uninfluenced by
force, moving round the surface of an anchor-ring, along either the
outer great circle, or along a sinuous geodetic such as we have con-
sidered in § 311, in which clearly there are an infinite number of foci
each conjugate to any one point of the path, at equal successive dis-
tances from one another.
317. If/- I distinct^ courses from a configuration (9, each difl'ering
infinitely little from a certain natural course O ..E . . (9, .. O ^
<^i-i- • <2j cut it in configurations Oj, O^, O^, . . . 0^_^, and if, besides
these, there are not on it any other kinetic foci conjugate to (9,
between O and ft and no focus at all, conjugate to E^ between
E and Q, the action in this natural course from O to Q \^ the
maximum for all courses 0.,.P^, P^...Q; P^ being a configura-
* In our second volume we hope to give all necessary elementary explanations on
this subject.
2 Two courses are here called not distinct if they differ from one another only in
the absolute magnitude, not in the proportions, of the components of the deviations
by which they differ from the standard course.
no PRELIMINARY.
tion infinitely nearly agreeing with some configuration between E
and O^ of the standard course O . . E .. (9, . . O^ O^^^ . . Q,
and O . .. R^, P,-" Q denoting the natural courses between O and R^,
and R^ and Q, which deviate infinitely little from this standard course.
318. Considering now, for simplicity, only cases in which there
are but two degrees (§ 165) of freedom to move, we see that after
any infinitely small conservative disturbance of a system in passing
through a certain configuration, the system will first again pass
through a configuration of the undisturbed course, at the first con-
figuration of the latter at which the action in the undisturbed motion
ceases to be a minimum. For instance, in the case of a particle,
confined to a surface, and subject to any conservative system of force,
an infinitely small conservative disturbance of its motion through any
point, O, produces a disturbed path, which cuts the undisturbed path
at the first point, 0\ at which the action in the undisturbed path from
O ceases to be a minimum. Or, if projectiles, under the influence of
gravity alone, be thrown from one point, O, in all directions with
equal velocities, in one vertical plane, their paths, as is easily proved,
intersect one another consecutively in a parabola, of which the focus
is (9, and the vertex the point reached by the particle projected
directly upwards. The actual course of each particle from O is the
course of least possible action to any point, R, reached before the
enveloping parabola, but is not a course of minimum action to any
point, Q, in its path after the envelop is passed.
319. Or again, if a particle slides round along the greatest circle of
the smooth inner surface of a hollow anchor-ring, the 'action,' or
simply the length of path, from point to point, will be least possible
for lengths (§ 305) less than tt Jab. Thus if a string be tied round
outside on the greatest circle of a perfectly smooth anchor-ring, it will
slip off unless held in position by staples, or checks of some kind, at
distances of not less than this amount, Trjab, from one another in
succession round the circle. With reference to this example, see also
§ 314, above.
Or, if a particle slides down an inclined hollow cylinder, the
action from any point will be the least possible along the straight path
to any other point reached in a time less than that of the vibration
one way of a simple pendulum of length equal to the radius of the
cylinder, and influenced by a force equal to g cos /, instead of g the
whole force of gravity. But the action will not be a minimum from
any point, along the straight path, to any other point reached in a
longer time than this. The case in which the groove is horizontal
(i = o) and the particle is projected along it, is particularly simple and
instructive, and may be worked out in detail with great ease, without
assuming any of the general theorems regarding action.
CHAPTER III.
EXPERIENCE.
320. By the term Experience, in physical science, we designate,
according to a suggestion of Herschel's, our means of becoming
acquainted with the material universe and the laws which regulate it.
In general the actions which we see ever taking place around us are
complex, or due to the simultaneous action of many causes. When,
as in astronomy, we endeavour to ascertain these causes by simply
watching their effects, we observe; when, as in our laboratories, we
interfere arbitrarily with the causes or circumstances of a pheno-
menon, we are said to experiment.
321. For instance, supposing that we are possessed of instru-
mental means of measuring time and angles, we may trace out by
successive observations the relative position of the sun and earth at
different instants; and (the method is not susceptible of any accuracy,
but is alluded to here only for the sake of illustration) from the
variations in the apparent diameter of the former we may calculate
the ratios of our distances from it at those instants. We have thus a
set of observations involving time, angular position with reference to
the sun, and ratios of distances from it ; sufficient (if numerous
enough) to enable us to discover the laws which connect the varia-
tions of these co-ordinates.
Similar methods may be imagined as applicable to the motion of
any planet about the sun, of a satellite about its primary, or of one
star about another in a binary group.
322. In general all the data of Astronomy are determined in this
way, and the same may be said of such subjects as Tides and Meteor-
ology. Isothermal Lines, Lines of Equal Dip or Intensity, Lines of
No Declination, the Connexion of Solar Spots with Terrestrial Mag-
netism, and a host of other data and phenomena, to be explained
under the proper heads in the course of the work, are thus deducible
from Observation merely. In these cases the apparatus for the gigantic
experiments is found ready arranged in Nature, and all that the
philosopher has to do is to watch and measure their progress to its
last details.
1 1 2 PRELIMINAR V.
323. Even in the instance we have chosen above, that of the
planetary motions, the observed eifects are complex ; because, unless
possibly in the case of a double star, we have no instance of the
widistiirbed action of one heavenly body on another; but to a first
approximation the motion of a planet about the sun is found to be
the same as if no other bodies than these two existed; and the
approximation is sufficient to indicate the probable law of mutual
action, whose full confirmation is obtained when, its truth being
assumed, the disturbing effects thus calculated are allowed for, and
found to account completely for the observed deviations from the
consequences of the first supposition. This may serve to give an
idea of the mode of obtaining the laws of phenomena, which can
only be observed in a complex form; and the method can always be
directly applied when one cause is known to be pre-eminent.
324. Let us take a case of the other kind — that in which the effects
are so complex that we cannot deduce the causes from the observation
of combinations arranged in Nature, but must endeavour to form for
ourselves other combinations which may enable us to study the effects
of each cause separately, or at least with only slight modification from
the interference of other causes.
A stone, when dropped, falls to the ground; a brick and a boulder,
if dropped from the top of a cliff at the same moment, fall side by
side, and reach the ground together. But a brick and a slate do not;
and while the former falls in a nearly vertical direction, the latter
describes a most complex path. A sheet of paper or a fragment of
gold-leaf presents even greater irregularities than the slate. But by
a slight modification of the circumstances, we gam a considerable
insight into the nature of the question. The paper and gold-leaf, if
rolled into balls, fall nearly in a vertical line. Here, then, there are
evidently at least two causes at work, one which tends to make all
bodies fall, and that vertically; and another which depends on the
form and substance of the body, and tends to retard its fall and alter
its vertical direction. How can we study the effects of the former on
all bodies without sensible complication from the latter? The effects
of Wind, etc., at once point out what the latter cause is, the air (whose
existence we may indeed suppose to have been discovered by such
effects); and to study the nature of the action of the former it is
necessary to get rid of the complications arising from the presence
of air. Hence the necessity for Experhtient. By means of an appa-
ratus to be afterwards described, we remove the greater part of the
air from the interior of a vessel, and in that we try again our expe-
riments on the fall of bodies; and now a general law, simple in the
extreme, though most important in its consequences, is at once appa-
rent— viz. that all bodies, of whatever size, shape, or material, if
dropped side by side at the same instant, fall side by side in a space
void of air. Before experiment had thus separated the phenomena,
hasty philosophers had rushed to the conclusion that some bodies
possess the quality of heaviness^ others that of lightness^ etc Had
EXPERIENCE. 113
this state of things remained, the law of gravitation, vigorous though
its action be throughout the universe, could never have been recog-
nized as a general principle by the human mind.
Mere observation of lightning and its effects could neVer have led
to the discovery of their relation to the phenomena presented by
rubbed amber. A modification of the course of Nature, such as the
bringing down of atmospheric electricity into our laboratories, was
necessary. Without experiment we could never even have learned
the existence of terrestrial magnetism.
325. When a particular agent or cause is to be studied, experi-
ments should be arranged in such a way as to lead if possible
to results depending on it alone; or, if this cannot be done, they
should be arranged so as to show differences produced by vary-
ing it.
326. Thus to determine the resistance of a wire against the
conduction of electricity through it, we may measure the whole
strength of current produced in it by electromotive force between
its ends when the amount of this electromotive force is give7t,
or can be ascertained. But when the wire is that of a submarine
telegraph cable there is always an tmknown and ever varying
electromotive force between its ends, due to the earth (produc-
ing what is commonly called the "earth-current"), and to deter-
mine its resistance the difference in the strength of the current
produced by suddenly adding to or subtracting from the terres-
trial electromotive force, the electromotive force of a given
voltaic battery is to be very quickly measured; and this is to be
done over and over again, to eliminate the effect of variation of
the earth current during the i^^ seconds of time which must
elapse before the electro- static induction permits the current due to
the battery to reach nearly enough its full strength to practically
annul error on this score.
327. Endless patience and perseverance in designing and trying
different methods for investigation are necessary for the advancement
of science : and indeed, in discovery, he is the most likely to succeed
who, not allowing himself to be disheartened by the non-success of
one form of experiment, judiciously varies his methods, and thus
interrogates in every conceivably useful manner the subject of his
investigations.
328. A most important remark, due to Herschel, regards what are
called r^^V///^/ phenomena. When, in an experiment, all known causes
being allowed for, there remain certain unexplained effects (exces-
sively slight it may be), these must be carefully investigated, and every
conceivable variation of arrangement of apparatus, etc., tried; until, if
possible, we manage so to exaggerate the residual phenomenon as to
be able to detect its cause. It is here, perhaps, that in the present
state of science we may most reasonably look for extensions of our
knowledge ; at all events we are warranted by the recent history of
Natural Philosophy in so doing. Thus, to take only a very few
T. 8
1 14 PRELIMINARY.
instances, and to say nothing of the discovery of electricity and mag-
netism by the ancients, the pecuUar smell observed in a room in
which an electrical machine is kept in action, was long ago observed,
but called the 'smell of electricity,' and thus left unexplained. The
sagacity of Schonbein led to the discovery that this is due to the
formation of Ozone, a most extraordinary body, of enormous chem-
ical energies ; whose nature is still uncertain, though the attention of
chemists has for years been directed to it.
329. Slight anomalies in the motion of Uranus led Adams and
Le Verrier to the discovery of a new planet; and the fact that a
magnetized needle comes to rest sooner when vibrating above a
copper plate than when the latter is removed, led Arago to what
was once called magnetism of rotation, but has since been explained,
immensely extended, and applied to most important purposes. In
fact, this accidental remark about the oscillation of a needle led
to facts from which, in Faraday's hands, was evolved the grand
discovery of the Induction of Electrical Currents by magnets or
by other currents. We need not enlarge upon this point, as in
the following pages the proofs of the truth and usefulness of the
principle will continually recur. Our object has been not so much
to give applications as methods, and to show, if possible, how to
attack a new combination, with the view of separadng and studying
in detail the various causes which generally conspire to produce
observed phenomena, even those which are apparently the simplest.
330. If, on repetidon several times, an experiment continually gives
different results, it must either have been very carelessly performed,
or there must be some disturbing cause not taken account of. And,
on the other hand, in cases where no very great coincidence is
likely on repeated trials, an unexpected degree of agreement between
the results of various trials should be regarded with the utmost
suspicion, as probably due to some unnoticed peculiarity of the
apparatus employed. In either of these cases, however, careful
observation cannot fail to detect the cause of the discrepancies or
of the unexpected agreement, and may possibly lead to discoveries
in a totally unthought-of quarter. Instances of this kind may be
given without limit ; one or two must suffice.
331. Thus, with a very good achromatic telescope a star appears
to have a sensible disc. But, as it is observed that the discs of
all stars appear to be of equal angular diameter, we of course suspect
some comrnon error. Limiting the aperture of the object-glass
increases the appearance in question, which, on full investigation,
is found to have nothing to do with discs at all. It is, in fact, a dif-
fraction phenomenon, and will be explained in our chapters on Light.
Again, in measuring the velocity of Sound by experiments con-
ducted at night with cannon, the results at one station were never
found to agree exactly with those at the other ; sometimes, indeed,
the differences were very considerable. But a little consideration
EXPERIENCE. 1 1 5
led to the remark, that on those nights in which the discordance
was greatest a strong wind was blowing nearly from one station
to the other. Allowing for the obvious effect of this, or rather
eliminating it altogether, the mean velocities on different evenings
were found to agree very closely.
332. It may perhaps be advisable to say a few words here about
the use of hypotheses, and especially those of very different gradations
of value which are promulgated in the form of Mathematical Theories
of different branches of Natural Philosophy.
333. Where, as in the case of the planetary motions and disturb-
ances, the forces concerned are thoroughly known, the mathematical
theory is absolutely true, and requires only analysis to work out its
remotest details. It is thus, in general, far ahead of observation, and
is competent to predict effects not yet even observed — as, for instance,
Lunar Inequalities due to the action of Venus upon the Earth, etc. etc.,
to which no amount of observation, unaided by theory, would ever
have enabled us to assign the true cause. It may also, in such
subjects as Geometrical Optics, be carried to developments far beyond
the reach of experiment ; but in this science the assumed bases of the
theory are only approximate, and it fails to explain in all their peculi-
arities even such comparatively simple phenomena as Halos and
Rainbows ; though it is perfectly successful for the practical purposes
of the maker of microscopes and telescopes, and has, in these cases,
carried the construction of instruments to a degree of perfection
which merely tentative processes never could have reached.
334. Another class of mathematical theories, based to a certain
extent on experiment, is at present useful, and has even in certain
cases pointed to new and important results, which experiment has
subsequently verified. Such are the Dynamical Theory of Heat, the
Undulatory Theory of Light, etc. etc. In the former, which is based
upon the experimental fact that heat is motion^ many formulae are
at present obscure and uninterpretable, because we do not know
what is moving or how it moves. Results of the theory in which
these are not involved, are of course experimentally verified. The
same difficuldes exist in the Theory of Light. But before this
obscurity can be perfectly cleared up, we must know something
of the ultimate, or molecular, constitution of the bodies, or groups
of molecules, at present known to us only in the aggregate.
335. A third class is well represented by the Mathematical Theories
of Heat (Conduction), Electricity (Statical), and Magnetism (Perma-
nent). Although we do not know how Heat is propagated in bodies,
nor what Statical Electricity or Permanent Magnetism are, the laws
of their forces are as certainly known as that of Gravitation, and
can therefore like it be developed to their consequences, by the
application of Mathematical Analysis. The works of Fourier',
^ Theorie Ajtalytique de la Chaleur. Paris, 1822.
8—2
1 1 6 PRELIMINAR V.
Green \ and Poisson'', are remarkable instances of such develop-
ment. Another good example is Ampere's Theory of Electro-
dynamics.
336. Mathematical theories of physical forces are, in general, of
one of two species. First, those in which the fundamental assump-
tion is far more general than is necessary. Thus the celebrated
equation of Laplace's Functions contains the mathematical foundation
of the theories of Gravitation, Statical Electricity, Permanent Mag-
netism, Permanent Flux of Heat, Motion of Incompressible Fluids,
etc. etc., and has therefore to be accompanied by limiting consider-
ations when applied to any one of these subjects.
337. Again, there are those which are built upon a few experiments,
or simple but inexact hypotheses, only; and which require to be
modified in the way of extension rather than limitation. As a notable
example, we may refer to the whole subject of Abstract Dynamics,
which requires extensive modifications (explained in Division III.)
before it can, in general, be applied to practical purposes.
338. When the most probable result is required from a number of
observations of the same quantity which do not exactly agree, we
must appeal to the mathematical theory of probabilities to guide us
to a method of combining the results of experience, so as to eliminate
from them, as far as possible, the inaccuracies of observation. But
it must be explained that we do not at present class as inaccuracies
of ohsei'vation any errors which may affect alike every one of a series
of observations, such as the inexact determination of a zero-point or
of the essential units of time and space, the personal equation of the
observer, etc. The process, whatever it may be, which is to be
employed in the elimination of errors, is applicable even to these, but
only when several distinct series of observations have been made, with
a change of instrument, or of observer, or of both.
339. We understand as inaccuracies of observation the whole class
of errors which are as likely to lie in one direction as another in suc-
cessive trials, and which we may fairly presume would, on the average
of an infinite number of repetitions, exactly balance each other in
excess and defect. Moreover, we consider only errors of such a
kind that their probability is the less the greater they are ; so that
such errors as an accidental reading of a wrong number of whole
degrees on a divided circle (which, by the way, can in general be
probably corrected by comparison with other observations) are not to
be included.
340. Mathematically considered, the subject is by no means an
easy one, and many high authorities have asserted that the reasoning
employed by Laplace, Gauss, and others, is not well founded ; although
the results of their analysis have been generally accepted. As an
excellent treatise on the subject has recently been published by Airy,
1 Essay on the Application of Mathematical Analysis to the Theories oj
Electricity and Magnetism. Nottingham, 1828. KidY^vmi^A in Crelle's Journal .
2 M6 moires snr le Magneiisme. Mem. de VAcad, des Sciences, 181 r.
EXPERIENCE. 117
it is not necessary for us to do more than sketch in the most cursory
manner what is called the Method of Least Squares.
341. Supposing the zero-point and the graduation of an instrument
(micrometer, mural circle, thermometer, electrometer, galvanometer,
etc.) to be absolutely accurate, successive readings of the value of a
quantity (linear distance, altitude of a star, temperature, potential,
strength of an electric current, etc.) may, and in general do, con
tinually differ. What is most probably the true value of the observed
quantity ?
The most probable value, in all such cases, if the observations are all
equally reliable, will evidently be the simple mean ; or if they are not
equally reliable, the mean found by attributing weights to the several
observations in proportion to their presumed exactness. But if several
such means have been taken, or several single observations, and if
these several means or observations have been differently qualified
for the determination of the sought quantity (some of them being
likely to give a more exact value than others), we must assign theoret-
ically the best method of combining them in practice.
342. Inaccuracies of observation are, in general, as likely to be in
excess as in defect. They are also (as before observed) more likely
to be small than great ; and (practically) large errors are not to be
expected at all, as such would come under the class of avoidable mis-
takes. It follows that in any one of a series of observations of the
same quantity the probability of an error of magnitude x, must depend
upon x^, and must be expressed by some function whose value
diminishes very rapidly as x increases. The probability that the
error lies between x and x-v'^x^ where Ix is very small, must also be
proportional to Sx The law of error thus found is
I -^nx
Jtt h
where ^ is a constant, indicating the degree of coarseness or delicacy
of the system of measurement employed. The co-efficient —j~ secures
that the sum of the probabilities of all possible errors shall be unity,
as it ought to be.
343. The Probable Error of an observation is a numerical quantity
such that the error of the observation is as Ukely to exceed as to fall
short of it in magnitude.
If we assume the law of error just found, and call P the probable
error in one trial, we have the approximate result
p= 0-477/2.
344. The probable error of any given multiple of the value of an
observed quantity is evidently the same multiple of the probable error
of the quantity itself
The probable error of the sum or difference of two quantities,
affected by independent errors, is the square root of the sum of the
squares of their separate probable errors.
1 1 8 PRELIMINAR V,
345. As above remarked, the principal use of this theory is in the
deduction, from a large series of observations, of the values of the
quantities sought in such a form as to be liable to the smallest pro-
bable error. As an instance — by the principles of physical astronomy,
the place of a planet is calculated from assumed values of the elements
of its orbit, and tabulated in the Nautical Almanac. The observed
places do not exactly agree with the predicted places, for two reasons
■ — first, the data for calculation are not exact (and in fact the main
object of the observation is to correct their assumed values); second,
the observation is in error to some unknown amount. Now the
difference between the observed, and the calculated, places depends
on the errors of assumed elements and of observation. Our methods
are applied to eliminate as far as possible the second of these, and the
resulting equations give the required corrections of the elements.
Thus if B be the calculated R. A. of a planet : ha, Be, 8^, etc., the
corrections required for the assumed elements : the true R.A. is
6 + ASa + £Se + nSra- + etc.,
where A, E, 11, etc., are approximately known. Suppose the observed
R.A. to be ©, then
e + ABa + EBe + UBzn +... = ©,
or ABa + EBe + HSra- + . . . = © - ^,
a known quantity, subject to error of observation. Every observation
made gives us an equation of the same form as this, and in general
the number of observations greatly exceeds that of the quantities Ba,
Be, Bw, etc., to be found.
346. The theorems of § 344 lead to the following rule for com-
bining any number of such equations which contain a smaller number
of unknown quantities : —
Make the probable error of the second member the same in each equa-
tion, by the employ me^tt of a proper factor : midtiply each equation by the
co-efficie?it of x in it and add all, for one of the final equations ; and so,
with reference to y, z, etc., for the others. The probable errors of the
values of x, y, etc., found from these final equations will be less than
those of the values derived from any other linear method of com-
bining the equations.
This process has been called the method of Least Squares, because
the values of the unknown quantities found by it are such as to render
the sum of the squares of the errors of the original equations a
minimum.
347. When a series of observations of the same quantity has been
made at different times, or under different circumstances, the law
connecting the value of the quantity with the time, or some other
variable, may be derived from the results in several ways — all more
or less approximate. Two of these methods, however, are so much
more extensively used than the others, that we shall devote a page or
EXPERIENCE.
119
two here to a preliminary notice of them, leaving detailed instances
of their application till we come to Heat, Electricity, etc. They
consist in (i) a Curve, giving a graphic representation of the relation
between the ordinate and abscissa, and (2) an Empirical Formula
connecting the variables.
348. Thus if the abscissae represent intervals of time, and the
ordinates the corresponding height of the barometer, we may con-
struct curves which show at a glance the dependence of barometric
pressure upon the time of day; and so on. Such curves may be
accurately drawn by photographic processes on a sheet of sensitive
paper placed behind the mercurial column, and made to move past
it with a uniform horizontal velocity by clockwork. A similar pro-
cess is applied to the Temperature and Electricity of the atmosphere,
and to the components of terrestrial magnetism.
349. When the observations are not, as in the last section, con-
tinuous, they give us only a series of points in the curve, from which,
however, we may in general approximate very closely to the result
of continuous observation by drawing, libera manii, a curve passing
through these points. This process, however, must be employed
with great caution ; because, unless the observations are sufficiently
close to each other, most important fluctuations in the curve may
escape notice. It is applicable, with abundant accuracy, to all cases
where the quantity observed changes very slowly. Thus, for instance,
weekly observations of the temperature at depths of from 6 to 24 feet
underground were found by Forbes sufficient for a very accurate
approximation to the law of the phenomenon.
350. As an instance of the processes employed for obtaining an
empirical formula, we may mention methods of Interpolation, to which
the problem can always be reduced. Thus from sextant observations,
. at known intervals, of the altitude of the sun, it is a common problem
of Astronomy to determine at what instant the altitude is greatest,
and what is that greatest altitude. The first enables us to find the
true solar time at the place, and the second, by the help of the
Nautical Almanac, gives the latitude. The calculus of finite differ-
ences gives us formulae proper for various data ; and Lagrange has
shown how to obtain a very useful one by elementary algebra.
In finite differences we have
f{x + h) =f{x) + hXf(x) + ^-^~> AYix) + . . .
This is useful, inasmuch as the successive differences, ^/(x\
Ay(^), etc., are easily calculated from the tabulated results of obser-
vation, provided these have been taken for equal successive in-
crements of ^.
If for values x^, x^,...Xn, a function takes the values y^, y^, y^,...
y„, Lagrange gives for it the obvious expression
1 20 PRELIMINAR V.
Here is assumed that the function required is a rational and
integral one in x of the n-i^^ degree; and, in general, a similar
limitation is in practice applied to the other formula above ; for in
order to find the complete expression for /{x), it is necessary to
determine the values of A/(x), ^y{x), .... If ;2 of the co-efficients be
required, so as to give the n chief terms of the general value of/{x)f
we must have n observed simultaneous values of x and/(^), and the
expression becomes determinate and of the n—i^^ degree in ^.
In practice it is usually sufficient to employ at most three terms
of the first series. Thus to express the length / of a rod of metal as
depending on its temperature f, we may assume
l^ being the measured length at any temperature t^, A and B are to
be found by the method of least squares from values of / observed for
different given values of t.
351. These formulae are practically useful for calculating the
probable values of any observed element, for values of the in-
dependent variable lying within the range for which observation has
given values of the element. But except for values of the inde-
pendent variable either actually within this range, or not far beyond
it in either direction, these formulae express functions which, in
general, will differ more and more widely from the truth the further
their application is pushed beyond the range of observation.
In a large class of investigations the observed element is in its
nature a periodic function of the independent variable. The har-
monic analysis (§ 88) is suitable for all such. When the values of the
independent variable for which the element has been observed are
not equidiflferent the co-efficients, determined according to the method
of least squares, are found by a process which is necessarily very
laborious ; but when they are equidifferent, and especially when the
difference is a submultiple of the period, the equation derived from
the method of least squares becomes greatly simplified. Thus, if Q
denote an angle increasing in proportion to /, the time, through four
right angles in the period, T^ of the phenomenon; so that
let /(^) = ^^ + ^,cos^-i-^2Cos 2^+ ...
+ B^ sin ^ -I- ^2 sin 2^ + . . .
where A^^ A^, Ag,...B^, B^,... are unknown co-efficients, to be
determined so that /(O) may express the most probable value of the
element, not merely at times between observations, but through all
time as long as the phenomenon is strictly periodic. By taking as
many of these coefficients as there are of distinct data by observation,
the formula is made to agree precisely with these data. But in most
applications of the method, the periodically recurring part of the phe-
nomenon is expressible by. a small number of terms of the harmonic
series, and the higher terms, calculated from a great number of data,
EXPERIENCE. 121
express either irregularities of the phenomenon not likely to recur,
or errors of observation. Thus a comparatively small number of
terms may give values of the element even for the very times of ob-
servation, more probable than the values actually recorded as having
been observed, if the observations are numerous but not minutely
accurate.
The student may exercise himself in writing out the equations to
determine five, or seven, or more of the coefficients according to the
method of least squares ; and reducing them by proper formulae of
analytical trigonometry to their simplest and most easily calculated
forms where the values of Q for which f{Q) is given are equidifferent.
He will thus see that when the difference is -v-, i being any integer,
and when the number of the data is / or any multiple of it, the equa-
tions contain each of them only one of the unknown quantities : so
that the method of least squares affords the most probable values of
the co-efficients, by the easiest and most direct elimination.
OF THE
VNIVERSltY
CHAPTER IV.
MEASURES AND INSTRUMENTS.
352. Having seen in the preceding chapter that for the investiga-
tion of the laws of nature we must carefully watch experiments, either
those gigantic ones which the universe furnishes, or others devised
and executed by man for special objects — and having seen that in
all such observations accurate measurements of Time, Space, Force,
etc., are absolutely necessary, we may now appropriately describe a
few of the more useful of the instruments employed for these pur-
poses, and the various standards or units which are employed in
them.
353. Before going into detail we may give a rapid resiwie of the
principal Standards and Instruments to be described in this chapter.
As most, if not all, of them depend on physical principles to be
detailed in the course of this work, we shall assume in anticipation
the establishment of such principles, giving references to the future
division or chapter in which the experimental demonstrations are
more particularly explained. This course will entail a slight, but
unavoidable, confusion — slight, because Clocks, Balances, Screws,
etc., are familiar even to those who know nothing of Natural Phi-
losophy ; unavoidable, because it is in the very nature of our subject
that no one part can grow alone, each requiring for its full develop-
ment the utmost resources of all the others. But if one of our
departments thus borrows from others, it is satisfactory to find that
it more than repays by the power which its improvement affords
them.
354. We may divide our more important and fundamental instru-
ments into four classes —
Those for measuring Time ;
„ „ Space, linear or angular ;
Force;
„ „ Mass.
Other instruments, adapted for special purposes such as the
measurement of Temperature, Light, Electric Currents, etc., will
come more naturally under the head of the particular physical
energies to whose measurement they are applicable. Descriptions of
MEASURES AND INSTRUMENTS. 123
self-recording instruments such as tide-gauges, and barometers, ther-
mometers, electrometers, recording photographically or otherwise the
continuously varying pressure, temperature, moisture, electric poten-
tial of the atmosphere, and magnetometers recording photographi-
cally the continuously varying direction and magnitude of the terres-
trial magnetic force, must likewise be kept for their proper places in
our work.
Calculating Machines have also important uses in assisting
physical research in 'a great variety of ways. They belong to two
classes : —
I. Purely Arithmetical, dealing with integral numbers of units.
All of this class are evolved from the primitive use of the calculuses
or little stones for counters (from which are derived the very names
calculation and " The Calculus "), through such mechanism as that
of the Chinese Abacus, still serving its original purpose well in
infant schools, up to the Arithmometer of Thomas of Colmar and the
grand but partially realized conceptions of calculating machines by
Babbage.
II. Continuous Calculating Machines. These are not only useful
as auxiliaries for physical research but also involve important dy-
namical and kinematical principles belonging properly to our subject.
355. We shall now consider in order the more prominent instru-
ments of each of these four classes, and some of their most important
applications : —
Clock, Chronometer, Chronoscope, Applications to Observation
and to self-registering Instruments.
Vernier and Screw-Micrometer, Cathetometer, Spherometer,
Dividing Engine, Theodolite, Sextant or Circle.
Common Balance, Bifilar Balance, Torsion Balance, Pendulum,
Dynamometer.
Among Standards we may mention —
1. Time. — Day, Hour, Minute, Second, sidereal and solar.
2. Space. — Yard and Metre: Radian, Degree, Minute, Second.
3. Force. — Weight of a Pound or Kilogramme, etc., in any par-
ticular locality (gravitation unit); poundal or dyne. Kinetic
Unit.
4. Mass. — Pound, Kilogramme, etc.
356. Although without instruments it is impossible to procure or
apply any standard, yet, as without the standards no instrument could
give us absolute measure, we may consider the standards first -^
referring to the instruments as if we already knew their principles
and applications.
357. First we may notice the standards or units of angular
measure :
1 24 PRELIMINAR V.
Radian^ or angle whose arc is equal to radius ;
Degree, or ninetieth part of a right angle, and its successive
subdivisions into sixtieths called Minutes^ Seconds, Thirds, etc. The
division of the right angle into 90 degrees is convenient because it
makes the half-angle of an equilateral triangle (sin"' |) an integral
number (30) of degrees. It has long been universally adopted by all
Europe. The decimal division of the right angle, decreed by the
French Republic when it successfully introduced other more sweeping
changes, utterly and deservedly failed.
The division of the degree into 60 minutes and of the minute into
60 seconds is not convenient; and tables of the circular functions for
degrees and hundredths of the degree are much to be desired.
Meantime, when reckoning to tenths of a degree suffices for the
accuracy desired, in any case the ordinary tables suffice, as 6' is y^- of
a degree.
The decimal system is exclusively followed in reckoning by radians.
The value of two right angles in this reckoning is 3-14159... , or tt.
Thus IT radians is equal to 180". Hence i8o°-=-7ris 57°-29578 ... ,
or 57" 17' 44'^*8 is equal to one radian. In mathematical analysis,
angles are uniformly reckoned in terms of the radian.
358. The practical standard of time is the Siderial Day, being the
period, nearly constant', of the earth's rotation about its axis (§ 237).
From it is easily derived the Mean Solar Day, or the mean interval
which elapses between successive passages of the sun across the
meridian of any place. This is not so nearly as the former, an abso-
lute or invariable unit; secular changes in the period of the earth's
^ In our first edition of our larger treatise it was stated that Laplace had calculated
from ancient observations of eclipses that the period of the earth's rotation about
its axis had not altered by TTn^iJTrTT-irir of itself since 720 B.C. In § 830 it was
pointed out that this conclusion is overthrown by farther information from
Physical Astronomy acquired in the interval between the printing of the two
sections, in virtue of a correction which Adams had made as early as 1863 upon
Laplace's dynamical investigation of an acceleration of the moon's mean motion,
produced by the Sun's attraction, showing that only about half of the observed
acceleration of the moon's mean motion relatively to the angular velocity of the
earth's rotation was accounted for by this cause. [Quoting from the first edition,
§ 830.] "In 1859 Adams communicated to Delaunay his final result: — that at
'the end of a century the moon is 5" 7 before the position she would have,
'relatively to a meridian of the earth, according to the angular velocities of the
'two motions, at the beginning of the century, and the acceleration of the
'moon's motion truly calculated from the various disturbing causes then recog-
'nized. Delaunay soon after verified this result : and about the beginning of
'1866 suggested that the true explanation may be a retardation of the earth's
'rotation by tidal friction. Using this hypothesis, and allowing for the conse-
*quent retardation of the moon's mean motion by tidal reaction (§ 276), Adams,
'in an estimate which he has communicated to us, founded on the rough as-
* sumption that the parts of the earth's retardation due to solar and lunar tides
'are as the squares of the respective tide-generating forces, finds 22* as the
'error by which the earth would in a century get behind a perfect clock rated
' at the beginning of the century. If the retardation of rate giving this integral
'effect were uniform {§ 32), the earth, as a timekeeper, would be going slower
'by *22 of a second per year in the middle, or "44 of a second per year at the
'end, than at the beginning of a century."
MEASURES AND INSTRUMENTS, 125
revolution round the sun affect it, though very shghtly. It is divided
into 24 hours, and the hour, hke the degree, is subdivided into
successive sixtieths, called minutes and seconds. The usual sub-
division of seconds is decimal.
It is well to observe that seconds and minutes of time are distin-
guished from those of angular measure by notation. Thus we have
for time 13'' 43™ 27'-58, but for angular measure 13" 43' 2"]"-^^.
When long periods of time are to be measured, the mean solar
year, consisting of 366-242203 siderial days, or 365-242242 mean
solar days, or the century consisting of 100 such years, may be con-
veniently employed as the unit.
359. The ultimate standard of accurate chronometry must (if the
human race live on the earth for a few million years) be founded on
the physical properties of some body of more constant character
than the earth : for instance, a carefully-arranged metallic spring,
hermetically sealed in an exhausted glass vessel. The time of vibra-
tion of such a spring would be necessarily more constant from day to
day than that of the balance-spring of the best possible chronometer,
disturbed as this is by the train of mechanism with which it is con-
nected: and it would certainly be more constant from age to age
than the time of rotation of the earth, retarded as it now is by tidal
resistance to an extent that becomes very sensible in 2000 years;
and cooling and shrinking to an extent that must produce a very
considerable effect on its time-keeping in fifty million years.
360. The British standard of length is the Imperial Yard, defined
as the distance between two marks on a certain metallic bar, pre-
served in the Tower of London, when the whole has a temperature of
60° Fahrenheit. It was not directly derived from any fixed quantity
in nature, although some important relations wdth natural elements have
been measured with great accuracy. It has been carefully compared
with the length of a second's pendulum vibrating at a certain station in
the neighbourhood of London, so that should it again be destroyed,
as it was at the burning of the Houses of Parliament in 1834, and
should all exact copies of it, of which several are preserved in various
places, be also lost, it can be restored by pendulum observations. A
less accurate, but still (unless in the event of earthquake disturbance)
a very good, means of reproducing it exists in the measured base-lines
of the Ordnance Survey, and the thence calculated distances between
definite stations in the British Islands, which have been ascertained
in terms of it with a degree of accuracy sometimes within an inch
per mile, that is to say, within about g^o'^oTT*
361. In scientific investigations, we endeavour as much as possible
to keep to one unit at a time, and the foot, which is defined to be
one-third part of the yard, is, for British measurement, generally
adopted. Unfortunately the inch, or one-twelfth of a foot, must
sometimes be used, but it is subdivided decimally. The statute mile,
or 1760 yards, is unfortunately often used when great lengths on land
126 PRELIMINARY.
are considered; but the sea-mile, or average minute of latitude, is
much to be preferred. Thus it appears that the British measurement
of length is more inconvenient in its several denominations than the
European measurement of time, or angles.
362. In the French metrical system the decimal division is exclu-
sively employed. The standard, (unhappily) called the Metre, was
defined originally as the ten-millionth part of the length of the
quadrant of the earth's meridian from the pole to the equator ; but it
is now defined practically by the accurate standard metres laid up in
various national repositories in Europe. It is somewhat longer than
the yard, as the following Table shows :
Centimetre = '3937043 inch.
Metre = 3*280869 feet.
Kilometre = '6213767 British
Statute mile.
Inch =25'39977 millimetres.
Foot= 3*047972 decimetres.
British Statute mile
= 1609*329 metres.
363. The unit of superficial measure is in Britain the square yard,
in France the metre carre. Of course we may use square inches,
feet, or miles, as also square millimetres, kilometres, etc., or the
Hectare = 1 0,000 square mbtres.
Square inch= 6*451483 square centimetres.
„ foot= 9*290135 „ decimetres.
„ yard= 83*61121 „ decimetres.
Acre = '4046792 of a hectare.
Square British Statute mile = 258*9946 hectare.
Hectare = 2-471093 acres.
364. Similar remarks apply to the cubic measure in the two
countries, and we have the following Table : —
Cubic inch= 16*38661 cubic centimetres.
„ foot= 28*31606 „ decimetres or Z//r^j.
Gallon = 4-543808 litres.
„ =277-274 cubic inches, by Act of Parliament,
now repealed.
Litre = '0353 15 cubic feet.
365. The British unit of mass is the Pound (defined by standards
only) ; the French is the Kilogramme, defined originally as a litre of
water at its temperature of maximum density; but now practically
defined by existing standards.
Gramme = 15 '43 23 5 grains.
Kilogram. = 2*20362125 lbs.
Grain =64*79896 milligrammes.
Pound = 453*5927 grammes.
Professor W. H. Miller finds {Phil. Trans., 1857) that the 'kilo-
gramme des Archives^ is equal in mass to 15432*349 grains: and
the * kilogramme type laiton^ deposited in the Ministere de ITnterieure
in Paris, as standard for French commerce, is 15432*344 grains.
366. The measurement of force, whether in terms of the weight
of a stated mass in a stated locality, or in terms of the absolute or
MEASURES AND INSTRUMENTS. 127
kinetic unit, has been explained in Chapter II. (See §§221 — 227.)
From the measures of force and length we derive at once the measure
of work or mechanical effect. That practically employed by engi-
neers is founded on the gravitation measure of force. Neglecting the
difference of gravity at London and Paris, we see from the above
Tables that the following relations exist between the London and the
Parisian reckoning of work : —
Foot-pound =0-13825 kilogramme-metre.
Kilogramme-metre =7*2331 foot-pounds.
367. A Clock is primarily an instrument which, by means of a
train of wheels, records the number of vibrations executed by a
pendulum ; a Chronometer or Watch performs the same duty for the
oscillations of a flat spiral spring — ^just as the train of wheel-work in
a gas-meter counts the number of revolutions of the main shaft
caused by the passage of the gas through the machine. As, how-
ever, it is impossible to avoid friction, resistance of air, etc., a pendu-
lum or spring, left to itself, would not long continue its oscillations,
and, while its motion continued, would perform each oscillation in
less and less time as the arc of vibration diminished : a continuous
supply of energy is furnished by the descent of a weight, or the
uncoiling of a powerful spring. This is so applied, through the
train of wheels, to the pendulum or balance-wheel by means of a
mechanical contrivance called an Escapei?ie?it, that the oscillations are
maintained of nearly uniform extent, and therefore of nearly uniform
duration. The construction of escapements, as well as of trains of
clock-wheels, is a matter of Mechanics, with the details of which we
are not concerned, although it may easily be made the subject of
mathematical investigation. The means of avoiding errors intro-
duced by changes of temperature, which have been carried out in
Compensation pendulums and balances, will be more properly described
in our chapters on Heat. It is to be observed that there is little
inconvenience if a clock lose or gain regularly; that can be easily
and accurately allowed for : irregular rate is fatal.
368. By means of a recent application of electricity, to be after-
wards described, one good clock, carefully regulated from time to
time to agree with astronomical observations, may be made (without
injury to its own performance) to control any number of other less-
perfectly constructed clocks, so as to compel their pendulums to
vibrate, beat for beat, with its own.
369. In astronomical observations, time is estimated to tenths of
a second by a practised observer, who, while watching the phe-
nomena, counts the beats of the clock. But for the very accurate
measurement of short intervals, many instruments have been devised.
Thus if a small orifice be opened in a large and deep vessel full of
mercury, and if we know by trial the weight of metal that escapes
say in five minutes, a simple proportion gives the interval which
elapses during the escape of any given weight. It is easy to con-
1 28 PRELIMINAR V.
trive an adjustment by which a vessel may be placed under, and
withdrawn from, the issuing stream at the time of occurrence of any
two successive phenomena.
370. Other contrivances are sometimes employed, called Stop-
watches, Chronoscopes, etc., which can be read off at rest, started
on the occurrence of any phenomenon, and stopped at the oc-
currence of a second, then again read off; or which allow of the
making (by pressing a stud) a slight ink-mark, on a dial revolving
at a given rate, at the instant of the occurrence of each phe-
nomenon to be noted. But, of late, these have almost entirely given
place to the Electric Chronoscope, an instrument which will be fully
described later, when we shall have occasion to refer to experiments
in which it has been usefully employed.
371. We now come to the measurement of space, and of angles,
and for these purposes the most important instruments are the Vernier
and the Screw.
372. Elementary geometry, indeed, gives us the means of dividing
any straight line into any assignable number of equal parts; but in
practice this is by no means an accurate
or reliable method. It was formerly used
in the so-called Diagonal Scale, of which
the construction is evident from the dia-
gram. The reading is effected by a
sliding piece whose edge is perpendicular
to the length of the scale. Suppose that
it is PQ. whose position on the scale is
required. This can evidently cut only one
of the transverse lines. Its number gives
the number of tenths of an inch (4 in the
figure), and the horizontal line next above
the point of intersection gives evidently
the number of hundredths (in the present case 4). Hence the
reading is 7*44. As an idea of the comparative uselessness of
this method, we may mention that a quadrant of 3 feet radius,
which belonged to Napier of Merchiston, and is divided on the
limb by this method, reads to minutes of a degree; no higher
accuracy than is now attainable by the pocket sextants made by
Troughton and Simms, the radius of whose arc is virtually little
more than an inch. The latter instrument is read by the help
of a Vernier.
373. The Vernier is commonly employed for such instruments as
the Barometer, Sextant, and Cathetometer, while the Screw is applied
to the more delicate instruments, such as Astronomical Circles,
Micrometers, and the Spherometer.
374. The vernier consists of a slip of metal which slides
along a divided scale, the edges of the two being coincident.
Hence, when it is appUed to a divided circle, its edge is circular,
MEASURES AND INSTRUMENTS.
129
and it moves about an axis passing through the centre of the
divided Hmb.
In the sketch let 0, 1, 2, ... 10 denote the divisions on the vernier,
o, r, 2, etc., any set of consecutive divisions on the limb or scale
If, when 0 and o com-
Wv^AA,
^AA/^A/1
30
29-
■vutv^V^
^rO^
along whose edge it slides.
cide, 10 and n coincide also, then 10 divisions of
the vernier are equal in length to 11 on the limb;
and therefore each division of the vernier is yjths,
or ly^Q- of a division on the limb. If, then, the ver-
nier be moved till 1 coincides with i, 0 will be y^yth
of a division of the limb beyond o; if 2 coincide
with 2, 0 will be xV^''^ beyond o; and so on. Hence
to read the vernier in any position, note first the
division next to o, and behind it on the limb. This
is the integral number of divisions to be read. For
the fractional part, see which division of the vernier
is in a line with one on the limb; if it be the 4th
(as in the figure), that indicates an addition to the
reading of y^^ths of a division of the limb; and so on.
Thus, if the figure represent a barometer scale divided
in.
into inches and tenths, the reading is 30-34, the zero
line of the vernier being adjusted to the level of the mercury.
375. If the limb of a sextant be divided, as it usually is, to third-
parts of a degree, and the vernier be formed by dividing twenty-one
of these into twenty equal parts, the instrument can be read to
twentieths of divisions on the limb, that is, to minutes of arc.
If no Hne on the vernier coincide with one on the limb, then since
the divisions of the former are the longer there will be one of the
latter included between the two lines of the vernier, and it is usual
in practice to take the mean of the readings which would be given
by a coincidence of either pair of bounding lines.
376. In the above sketch and description, the numbers on the
scale and vernier have been supposed to run opposite ways. This
is generally the case with British instruments. In some foreign ones
the divisions run in the same direction on vernier and limb, and in
that case it is easy to see that to read to tenths of a scale division we
must have ten divisions of the vernier equal to 7iiiie of the scale.
In general to read to the ;/th part of a scale division, n divisions of
the vernier must equal n + \ or n- i divisions on the limb, according
as these run in opposite or similar directions.
377. The principle of the Scre7ef has been already noticed (§ 1 14).
It may be used in either of two ways, i.e. the nut may be fixed,
and the screw advance through it, or the screw may be prevented
from moving longitudinally by a fixed collar, in which case the nut,
if prevented by fixed guides from rotating, will move in the direction
of the common axis. The advance in either case is evidently pro-
portional to the angle through which the screw has turned about its .
T.
I30 PRELIMINARY.
axis, and this may be measured by means of a divided head fixed
perpendicularly to the screw at one end, the divisions being read oif
by a pointer or vernier attached to the frame of the instrument. The
nut carries with it either a tracing-point (as in the dividing engine) or
a wire, thread, or half the object-glass of a telescope (as in micro-
meters), the thread or wire, or the play of the tracing-point, being
at right angles to the axis of the screw.
378. Suppose it be required to divide a line into any number
of equal parts. The line is placed parallel to the axis of the screw
with one end exactly under the tracing-point, or under the fixed wire
of a microscope carried by the nut, and the screw-head is read off. By
turning the head, the tracing-point or microscope wire is brought to the
other extremity of the line; and the number of turns and fractions of
a turn required for the whole line is thus ascertained. Dividing this
by the number of equal parts required, we find at once the mnnber
of turns and fractional parts corresponding to one of the required
divisions, and by giving that amount of rotation to the screw over and
over again, drawing a Hne after each rotation, the required division is
effected.
379. In the Micrometer^ the movable wire carried by the nut
is parallel to a fixed wire. By bringing them into optical contact
the zero reading of the head is known; hence when another reading
has been obtained, we have by subtraction the number of turns
corresponding to the length of the object to be measured. The
absolute value of a turn of the screw is determined by calculation
from the number of threads in an inch, or by actually applying the
micrometer to an object of known dimensions.
380. For the measurement of the thickness of a plate, or the cur-
vature of a lens, the Spheroineter is used. It consists of a screw nut
rigidly fixed in the middle of a very rigid three-legged table, with its
axis perpendicular to the plane of the three feet (or finely rounded
ends of the legs,) and an accurately cut screw working in this nut.
The lower extremity of the screw is also finely rounded. The number
of turns, whole or fractional, of the screw, is read off by a divided
head and a pointer fixed to the stem. Suppose it be required to
measure the thickness of a plate of glass. The three feet of the
instrument are placed upon a nearly enough flat surface of a hard
body, and the screw is gradually turned until its point touches and
presses the surface. The muscular sense of touch perceives resistance to
the turning of the screw when, after touching the hard body, it presses
on it with a force somewhat exceeding the weight of the screw. The
first effect of the contact is a diminution of resistance to the turning,
due to the weight of the screw coming to be borne on its fine pointed
end instead of on the thread of the nut. The sudderi increase of
resistance at the instant when the screw commences to bear part of
the weight of the nut finds the sense prepared to perceive it with re-
markable delicacy on account of its contrast with the immediately
preceding diminution of resistance. The screw-head is now read off,
MEASURES AND INSTRUMENTS. 131
and the screw turned backwards until room is left for the insertion,
beneath its point, of the plate whose thickness is to be measured.
The screw is again turned until increase of resistance is again per-
ceived; and the screw-head is again read off. The difference of the
readings of the head is equal to the thickness of the plate, reckoned
in the proper unit of the screw and the division of its head.
381. If the curvature of a lens is to be measured, the instrument
is first placed, as before, on a plane surface, and the reading for the
contact is taken. The same operation is repeated on the spherical
surface. The difference of the screw readings is evidently the
greatest thickness of the glass which would be cut off by a plane
passing through the three feet. This is sufficient, with the distance
between each pair of feet, to enable us to calculate the radius of the
spherical surface.
In fact if a be the distance between each pair of feet, / the length
of screw corresponding to the difference of the two readings, R the
radius of the spherical surface; we have at once 2i? = — .+ /, or, as /
is generally very small compared with a^ the diameter is, very ap-
proximately, — ,.
3^
382. The Cathetometer is used for the accurate determination of
differences of level — for instance, in measuring the height to which a
fluid rises in a capillary tube above the exterior free surface. It
consists of a long divided metallic stem, turning round an axis as
nearly as may be parallel to its length, on a fixed tripod stand : and,
attached to the stem, a spirit-level. Upon the stem slides a metallic
piece bearing a telescope of which the length is approximately enough
perpendicular to the axis. The telescope tube is as nearly as may be
perpendicular to the length of the stem. By levelling screws in two
feet of the tripod the bubble of the spirit-level is brought to one
position of its glass when the stem is turned all round its axis. This
secures that the axis is vertical. In using the instrument the
telescope is directed in succession to the two objects whose difference
of level is to be found, and in each case moved (generally by a
delicate screw) up or down the stem, until a horizontal wire in the
focus of its eye-piece coincides with the image of the object. The
difference of readings on the vertical stem (each taken generally by
aid of a vernier sliding piece) corresponding to the two positions of
the telescope gives the required difference of level.
383. The principle of the Balance is generally known. We may
note here a few of the precautions adopted in the best balances to
guard against the various defects to which the instrument is liable ;
and the chief points to be attended to in its construction to secure
delicacy, and rapidity of weighing.
The balance-beam should be very stiff, and as light as possible
consistently with the requisite stiffness. For this purpose it is
9—2
132 PRELIMINARY.
generally formed either of tubes, or of a sort of lattice-fram6work.
To avoid friction, the axle consists of a knife-edge, as it is called ;
that is, a wedge of hard steel, which, when the balance is in use,
rests on horizontal plates of polished agate. A similar contrivance
is appHed in very delicate balances at the points of the beam from
which the scale-pans are suspended. When not in use, and just
before use, the beam with its knife-edge is lifted by a lever arrange-
ment from the agate plates. While thus secured it is loaded with
weights as nearly as possible equal (this can be attained by previous
trial with a coarser instrument), and the accurate determination is
then readily effected. The last fraction of the required weight is
determined by a rider, a very small weight, generally formed of wire,
which can be worked (by a lever) from the outside of the glass case
in which the balance is enclosed, and which may be placed in
different positions upon one arm of the beam. This arm is gra-
duated to tenths, etc., and thus shows at once the value of the rider in
any case as depending on its moment or leverage, § 233.
384. Qualities of a balance :
1. Stability. — For stability of the beam alone without pans and
weights, its centre of gravity must be below its bearing knife-edge.
For stability with the heaviest weights the line joining the points at
the ends of the beam from which the pans are hung must be below
the knife-edge bearing the whole.
2. Sensibility. — The beam should be sensibly deflected from a
horizontal position by the smallest difference between the weights in
the scale-pans. The definite measure of the sensibility is the angle
through which the beam is deflected by a stated difference between
the loads in the pans.
3. Quickness. — This means rapidity of oscillation, and consequently
speed in the performance of a weighing. It depends mainly upon the
depth of the centre of gravity of the whole below the knife-edge and
the length of the beam.
In our Chapter on Statics we shall give the investigation. The
sensibiHty and quickness are calculated for any given form and
dimensions of the instrument, in § 572.
A fine balance should turn with about a 500,000th of the greatest
load which can safely be placed in either pan.
The process of Double Weighings which consists in counterpoising a
mass by shot, or sand, or pieces of fine wire, and then substituting
weights for it in the same pan till equilibrium is attained, is more
laborious, but more accurate, than single weighing ; as it eliminates
all errors arising from unequal length of the arms, etc.
Correction is required for the weights of air displaced by the two
bodies weighed against one another when their difference is too large
to be negligable.
385. In the Torsion-balance invented, and used with great effect,
by Coulomb, a force is measured by the torsion of a fibre of silk, a
glass thread, or a metallic wire. The fibre- or wire is fixed at its
MEASURES AND INSTR UMENTS. 1 3 3
upper end, or at both ends, according to circumstances. In general
it carries a very light horizontal rod or needle, to the extremities of
which are attached the body on which is exerted the force to be
measured, and a counterpoise. The upper extremity of the torsion
fibre is fixed to an index passing through the centre of a divided
disc, so that the angle through which that extremity moves is directly
measured. If, at the same time, the angle through which the needle
has turned be measured, or, more simply, if the index be always
turned till the needle assumes a different position determined by
marks or sights attached to the case of the instrument — we have the
amount of torsion of the fibre, and it becomes a simple statical pro-
blem to determine from the latter the force to be measured ; its direc-
tion, and point of application, and the dimensions of the apparatus,
being known. The force of torsion as depending on the angle of
torsion was found by Coulomb to follow the law of simple proportion
up to the limits of perfect elasticity — as might have been expected
from Hooke's Law (see Properties of Matter), and it only remains
that we determine the amount for a particular angle in absolute
measure. This determination is, in general, simple enough in theory;
but in practice requires considerable care and nicety. The torsion-
balance, however, being chiefly used for comparative, not absolute,
measure, this determination is often unnecessary. More will be said
about it when we come to its application.
386. The ordinary spiral spring-balances used for roughly com-
paring either small or large weights or forces, are, properly speaking,
only a modified form of torsion-balance S as they act almost entirely
by the torsion of the wire, and not by longitudinal extension or by
flexure. Spring-balances we believe to be capable, if carefully con-
structed, of rivalling the ordinary balance in accuracy, while, for some
applications, they far surpass it in sensibility and convenience. They
measure directly force, not mass; and therefore if used for deter-
mining masses in diff"erent parts of the earth, a correction must be
applied for the varying force of gravity. The correction for tem-
perature must not be overlooked. These corrections may be avoided
by the method of double weighing.
387. Perhaps the most delicate of all instruments for the measure-
ment of force is the Pendulum. It is proved in Kinetics (see Div. II.)
that for any pendulum, whether oscillating about a mean vertical
position under the action of gravity, or in a horizontal plane, under
the action of magnetic force, or force of torsion, the square of the
number of small oscillations in a given time is proportional to the
magnitude of the force under which these oscillations take place.
For the estimation of the relative amounts of gravity at different
places, this is by far the most perfect instrument. The method of
coincidences by which this process has been rendered so excessively
delicate will be described later.
^ Binet. See also J. Thomson. Cambridge and Dublin Math. Journal, 1848.
1 34 PRELIMINAR Y.
In fact, the kinetic measure of force, as it is the first and
most truly elementary, is also far the most easy as well as perfect
method in many practical cases. It admits of an easy reduction to
gravitation measure.
388. Weber and Gauss, in constructing apparatus for observations
of terrestrial magnetism, endeavoured so to modify them as to admit
of their being read from some distance. For this purpose each bar,
made at that time too ponderous, carried a plain mirror. By means
of a scale, seen after reflection in the mirror and carefully read with
a telescope, it was of course easy to compute the deviations which the
mirror had experienced. But, for many reasons, it was deemed neces-
sary that the deflections, even under considerable force, should be
very small. With this view the Bifilar suspension was introduced.
The bar-magnet is suspended horizontally by two vertical wires or
fibres of equal length so adjusted as to share its weight equally
between them. When the bar turns, the suspension-fibres become
inclined to the vertical, and therefore the bar must rise. Hence, if
we neglect the torsion of the fibres, the bifilar actually measures a
force by comparing it with the weight of the suspended magnet.
Let a be the half length of the bar between the points of attach-
ment of the wires, ^ the angle through which the bar has been turned
(in a horizontal plane) from its position of equilibrium, / the length
of one of the wires.
Then if Q be the couple tending to turn the bar, and W its weight,
, ^ Wa^ sin^
we have O = — — . ,
2
which gives the couple in terms of the deflection 0.
If the torsion of the fibres be taken into account, it will be
sensibly equal to Q (since the greatest inclination to the vertical
is small), and therefore the couple resulting from it will be EO,
where E is some constant. This must be added to the value of Q,
just found in order to get the whole deflecting couple.
389. Ergometers are instruments for measuring energy. White's
friction brake measures the amount of work actually performed in
any time by an engine or other ' prime mover,' by allowing it during
the time of trial to waste all its work on friction. Morin!s ergometer
measures work without wasting any of it, in the course of its trans-
mission from the prime mover to machines in which it is usefully
employed. It consists of a simple arrangement of springs, measur-
ing at every instant the couple with which the prime mover turns the
shaft that transmits its work, and an integrating machine from which
the work done by this couple during any time can be read off.
390. White's friction brake consists of a lever clamped to the
shaft, but not allowed to turn with it. The moment of the force
required td prevent the lever from going round with the shaft,
sj ^--j. sin-
MEASURES AND INSTRUMENTS. isS
multiplied by the whole angle through which the shaft turns, measures
the whole work done against the friction of the clamp. The same
result is much more easily obtained by wrapping a rope or chain
several times round the shaft, or round a cylinder or drum carried
round by the shaft, and applying measured forces to its two ends
in proper directions to keep it nearly steady while the shaft turns
round without it. The difference of the moments of these two forces
round the axis, multiplied by the angle through which the shaft turns,
measures the whole work spent on friction against the rope. If we
remove all other resistance to the shaft, and apply the proper amount
of force at each end of the rope or chain (which is very easily done
in practice), the prime mover is kept running at the proper speed
for the test, and having its whole work thus wasted for the time and
measured.
DIVISION II.
ABSTRACT DYNAMICS.
CHAPTER v.— INTRODUCTORY.
391. Until we know thoroughly the nature of matter and the
forces which produce its motions, it will be utterly impossible to
submit to mathematical reasoning the exact conditions of any phy-
sical question. It has been long understood, however, that an ap-
proximate solution of almost any problem in the ordinary branches
of Natural Philosophy may be easily obtained by a species of ab-
straction^ or rather limitation of the data, such as enables us easily
to solve the modified form of the question, while we are well assured
that the circumstances (so modified) affect the result only in a super-
ficial manner.
392. Take, for instance, the very simple case of a crowbar em-
ployed to move a heavy mass. The accurate mathematical investi-
gation of the action would involve the simultaneous treatment of the
motions of every part of bar, fulcrum, and mass raised; and from our
almost complete ignorance of the nature of matter and molecular
forces, it is clear that such a treatment of the problem is impossible.
It is a result of observation that the particles of the bar, fulcrum,
and mass, separately, retain throughout the process nearly the same
relative positions. Hence the idea of solving, instead of the above
impossible question, another, in reality quite different, but, while
infinitely simpler, obviously leading to 7iearly the same results as the
former.
393. The new form is given at once by the experimental result
of the trial. Imagine the masses involved to be perfectly rigid (i.e.
incapable of changing their forms or dimensions), and the infinite
multiplicity of the forces, really acting, may be left out of consi-
deration ; so that the mathematical investigation deals with a finite
(and generally small) number of forces instead of a practically infinite
number. Our warrant for such a substitution is established thus.
ABSTRACT DYNAMICS. 137
394. The only effects of the intermolecular forces would be ex-
hibited in molecular alterations of the form or volume of the masses
involved. But as these (practically) remain almost unchanged, the
forces which produce, or tend to produce, changes in them may be
left out of consideration. Thus we are enabled to investigate the
action of machinery by supposing it to consist of separate portions
whose forms and dimensions are unalterable.
395. If we go a little farther into the question, we find that the
lever bends ^ some parts of it are extended and others compressed.
This would lead us into a very serious and difficult inquiry if we had
to take account of the whole circumstances. But (by experience) we
find that a sufficiently accurate solution of this more formidable case
of the problem may be obtained by supposing (what can fiever be
realized in practice) the mass to be homogeneous, and the forces
consequent on a dilatation, compression, or distortion, to be propor-
tional in magnitude, and opposed in direction, to these deformations
respectively. By this farther assumption, close approximations may
be made to the vibrations of rods, plates, etc., as well as to the statical
effects of springs, etc.
396. We may pursue the process farther. Compression, in general,
develops heat, and extension, cold. These alter sensibly the elas-
ticity of a body. By introducing such considerations, we reach,
without great difficulty, what may be called a third approximation
to the solution of the physical problem considered.
397. We might next introduce the conduction of the heat, so
produced, from point to point of the solid, with its accompanying
modifications of elasticity, and so on ; and we might then consider
the production of thermo-electric currents, which (as we shall see)
are always developed by unequal heating in a mass if it be not per-
fectly homogeneous. Enough, however, has been said to show,yf/'j/,
our utter ignorance as to the true and complete solution of any
physical question by the only perfect method, that of the consideration
of the circumstances which affect the motion of every portion, sepa-
rately, of each body concerned ; and, second, the practically sufficient
manner in which practical questions may be attacked by limiting their
generality, the li?7titations introduced being themselves deduced from ex-
perience, and being therefore Nature's own solution (to a less or
greater degree of accuracy) of the infinite additional number of
equations by which we should otherwise have been encumbered.
398. To take another case : in the consideration of the propa-
gation of waves on the surface of a fluid, it is impossible, not only
on account of mathematical difficulties, but on account of our igno-
rance of what matter is, and what forces its particles exert on each
other, to form the equations which would give us the separate motion
of each. Our first approximation to a solution, and one sufficient
for most practical purposes, is derived from the consideration of the
138 INTRODUCTORY,
motion of a homogeneous, incompressible, and perfectly plastic mass;
a hypothetical substance which, of course, nowhere exists in nature.
399. Looking a little more closely, we find that the actual motion
differs considerably from that given by the analytical solution of the
restricted problem, and we introduce farther considerations, such as
the comprssibility of fluids, their internal friction, the heat generated
by the latter, and its effects in dilating the mass, etc. etc. By such
successive corrections we attain, at length, to a mathematical result
which (at all events in the present state of experimental science)
agrees, within the limits of experimental error, with observation.
400. It would be easy to give many more instances substantiating
what has just been advanced, but it seems scarcely necessary to do
so. We may therefore at once say that there is no question in
physical science which can be completely and accurately investigated
by mathematical reasoning (in which, be it carefully remembered,
it is not necessary that symbols should be introduced), but that there
are different degrees of approximation, involving assumptions more
and more nearly coincident with observation, which may be arrived
at in the solution of any particular question.
401. The object of the present division of this work is to deal with the
first and secojid of these approxi^nations. In it we shall suppose all
solids either rigid, i.e. unchangeable in form and volume, or elastic;
but in the latter case, we shall assume the law, connecting a com-
pression or a distortion with the force which causes it, to have a
particular form deduced from experiment. And we shall also leave
out of consideration the thermal or electric effects which compression
or distortion generally produce. We shall also suppose fluids, whether
liquids or gases, to be either incompressible or compressible ac-
cording to certain known laws; and we shall omit considerations
of fluid friction, although we admit the consideration of friction
between solids. Fluids will therefore be supposed perfect^ i.e. such
that any particle may be moved amongst the others by the slightest
force.
402. When we come to Properties of Matter and the Physical
Forces, we shall give in detail, as far as they are yet known, the
modifications which farther approximations have introduced into the
previous results.
403. The laws of friction between solids were very ably investi-
gated by Coulomb; and, as we shall require them in the succeeding
chapters, we give a brief summary of them here ; reserving the more
careful scrutiny of experimental results to our chapter on Properties
of Matter.
404. To produce sliding of one solid body on another, the sur-
faces in contact being plane, requires a tangential force which
depends, — (i) upon the nature of the bodies; (2) upon their polish,
or the species and quantity of lubricant which may have been applied;
ABSTRACT DYNAMICS. 139
(3) upon the normal pressure between them, to which it is in general
directly proportional; (4) upon the length of time during which they
have been suffered to remain in contact.
It does not (except in extreme cases where scratching or abrasion
takes place) depend sensibly upon the area of the surfaces in contact.
This, which is called Statical Friction, is thus capable of opposing a
tangential resistance to motion which may be of any requisite amount
up to ^R\ where R is the whole normal pressure between the bodies;
and /x (which depends mainly upon the nature of the surfaces in
contact) is the co-efficient of Statical Friction. This co-efficient varies
greatly with the circumstances, being in some cases as low as ©'03, in
others as high as o'8o. Later we shall give a table of its values.
Where the applied forces are insufficient to produce motion, the
whole amount of statical friction is not called into play; its amount
then just reaches what is sufficient to equiUbrate the other forces, and
its direction is the opposite of that in which their resultant tends to
produce motion. When the statical friction has been overcome, and
sliding is produced, experiment shows that a force of friction con-
tinues to act, opposing the motion, sensibly proportional to the
normal pressure, and independent of the velocity. But for the same
two bodies the co-efficient of Kinetic Friction is less than that of Sta-
tical Friction, and is approximately the same whatever be the rate of
motion.
405. When among the forces acting in any case of equilibrium,
there are frictions of solids on solids, the circumstances would not
be altered by doing away with all friction, and replacing its forces by
forces of mutual action supposed to remain unchanged by any in-
finitely small relative motions of the parts between which they act.
By this artifice all such cases may be brought under the general
principle of Lagrange (§ 254).
406. In the following chapters on Abstract Dynamics we will
confine ourselves chiefly to such portions of this extensive subject
as are Hkely to be useful to us in the rest of the work.
CHAPTER VI.
STATICS OF A PARTICLE.— ATTRACTION.
407. We naturally divide Statics into two parts — the equilibrium
of a Particle^ and that of a rigid or elastic Body or System of Fariicles
whether solid or fluid. The second law of motion suffices for one
part — for the other, the third, and its consequences pointed out by
Newton, are necessary. In the succeeding sections we shall dispose
of the first of these parts, and the rest of this chapter will be devoted
to a digression on the important subject of Attraction.
408. By § 2 21, forces acting at the same point, or on the same
material particle, are to be compounded by the same laws as velo-
cities. Therefore the sum of their resolved parts in any direction
must vanish if there is equilibrium j whence the necessary and sufii-
cient conditions.
They follow also directly from Newton's statement with regard to
work, if we suppose the particle to have any velocity, constant in
direction and magnitude (and § 211, this is the most general sup-
position we can make, since absolute rest has for us no meaning).
For the work done in any time is the product of the displacement
during that time into the algebraic sum of the effective components
of the applied forces, and there is no change of kinetic energy.
Hence this sum must vanish for every direction. Practically, as any
displacement may be resolved into three, in any three directions not
coplanar, the vanishing of the work for any one such set of three
suffices for the criterion. But, in general, it is convenient to assume
them in directions at right angles to each other.
Hence, for the equilibrium of a material particle, it is necessary^ and
sufficient^ that the (algebraic) sums of the applied forces, resolved in
any one set of three rectangular directions, should vanish.
409. We proceed to give a detailed exposition of the results
which follow from the first clause of § 408. For three forces only we
have the following statement.
The resultant of two forces, acting on a material point, is repre-
STATICS OF A PARTICLE. -^ATTRACT! ON, 141
sented in direction and magnitude by the diagonal, through that
point, of the parallelogram described upon lines representing the
forces.
410. Parallelogram of forces stated symmetrically as to the three
forces concerned^ usually called the Triangle of Forces. If the lines
representing three forces acting on a material point be equal and
parallel to the sides of a triangle, and in directions similar to those
of the three sides when taken in order round the triangle, the three
forces are in equilibrium.
Let GEF be a triangle, and
let MA, MB, MC, be respectively
equal and parallel to the three
sides EF, FG, GE of this trian-
gle, and in directions similar to
the consecutive directions of
these sides in order. The point
Mis in equilibrium.
411. [True Triangle of Forces. Let three
tive directions round a triangle, DEE, and be
represented respectively by its sides : they are
not in equilibrium, but are equivalent to a
couple. To prove this, through D draw DH,
equal and parallel to EF, and in it introduce
a pair of balancing forces, each equal to EF.
Of the five forces, three, DE, DH and ED,
are in equilibrium, and may be removed ;
and there are then left two forces, EF and HD, equal, parallel, and
in dissimilar directions, which constitute a couple.]
412. To find the resultant of any number of forces in lines through
one point, not necessarily in one plane —
forces act in consecu-
Let MA^, MA^,MA.
MA.
^ repre-
sent four forces acting on M, in one
plane; required their resultant.
Find by the parallelogram of forces,
the resultant of two of the forces, MA^
and MA^. It will be represented by
MD'. Then similarly, find MD", the
resultant of MD' (the first subsidiary
resultant), and MA^, the third force.
Lastly, find MD'", the resultant of
MD" and MA^, MD" represents the
resultant of the given forces.
Thus, by successive applications of the fundamental proposition,
the resultant of any number of forces in lines through one point can
be found.
413. In executing this construction, it is not necessary to describe
142
ABSTRACT DYNAMICS.
the successive parallelograms, or even to draw their diagonals. It is
enough to draw through the given point
a line equal and parallel to the repre-
sentative of any one of the forces ;
through the point thus arrived at, to
draw a line equal and parallel to the
representative of another of the forces,
and so on till all the forces have been
taken into account. In this way we get
such a diagram as the annexed.
The several given forces may be taken
in any order, in the construction just
described. The resultant arrived at is
necessarily the same, whatever be the order in which we choose to
take them, as we may easily verify by elementary geometry.
In the fig. the order is MA^, ^^5>
MA^, MA^, MA^.
414. If, by drawing lines equal and
parallel to the representatives of the forces,
a closed figure is got, that is, if the line
last drawn leads us back to the point
from which we started, the forces are in
equilibrium. If, on the other hand, the
figure is not closed (§ 413), the resultant
is obtained by drawing a line from the
starting-point to the point finally reached;
(from M to T>) : and a force represented by DAf will equilibrate the
system.
415. Hence, in general, a set of forces represented by lines equal
and parallel to the sides of a complete polygon, are in equilibrium,
provided they act in lines through one point, in directions similar to
the directions followed in going round the polygon in one way.
416. Polygon of Forces. The construction we have just con-
sidered, is sometimes called the polygon of forces; but the true
polygon of forces, as we shall call it, is something quite different.
In it the forces are actually along the sides of a polygon, and repre-
sented by them in magnitude. Such a system must clearly have a
turning tendency, and it may be demonstrated to be reducible to one
couple.
417. In the preceding sections we have explained the principle
involved in finding the resultant of any number of forces. We have
now to exhibit a method, more easy than the parallelogram of forces
affords, for working it out in actual cases, and especially for obtaining
a convenient specification of the resultant. The instrument employed
for this purpose is Trigonometry.
418. A distinction may first be pointed out between two classes
of problems, direct and inverse. Direct problems are those in which
the resultant of forces is to be found ; inverse, those in which com-
STATICS OF A FARIYCLE.-^ATTR ACTION. 143
ponents of a force are to be found. The former class is fixed and
determinate ; the latter is quite indefinite, without limitations to be
stated for each problem. A system of forces can produce only one
effect; but an infinite number of systems can be obtained, which
shall produce the same effect as one force. The problem, therefore,
of finding components must be, in some way or other, limited, 'lliis
may be done by giving the lines along which the components are to
act. To find the components of a given force, in any three given
directions, is, in general, as we shall see, a perfectly determinate
problem.
Finding resultants is called Composition of Forces.
Finding components is called Resolution of Forces.
419. Co7?tpositio7i of Forces.
Required in position and magnitude the resultant of two given
forces acting in giving lines on a material point.
Let MA, MB represent two forces,
F and Q, acting on a material point M.
Let the angle BMA be denoted by i.
Required the magnitude of the resultant,
and its inclination to the line of either
force. M P
Let F denote the magnitude of the resultant; let a denote the
angle FMA, at which its line MD is inclined to MA, the line of the
first force F; and let ^ denote the angle DMB, at which it is inclined
to MB, the direction of the force (9.
Given F, Q, and t : required F, and a or p.
We have
MB' = MA"" + MB"" - 2MA.MB X cos MAD.
Hence, according to our present notation,
R'=.F' + Q'- 2FQ cos (180"- 0,
or F^ = F^+Q^ + 2FQco?>i.
Hence F ={F''+ Q' ^- 2FQC0?, i)K (i)
To determine a and ^ after the resultant has been found ; we have
sin DMA = -rp^ sin MAD,
MD '
or sma = ^smi, (2)
and similarly,
p
sin/3=-^sint. (s)
420. These formulae are useful for many applications ; but they
have the inconvenience that there may be ambiguity as to the angle,
whether it is to be acute or obtuse, which is to be taken when either
sin a or sin /8 has been calculated. If t is acute, both a and /? are
acute, and there is no ambiguity. If i is obtuse, one of the two
2MD . MA
or
^^^'^- 2RF^
and similarly,
144 ABSTRACT DYNAMICS.
angles, a, yS, might be either acute or obtuse ; but as they cannot be
both obtuse, the smaller of the two must, necessarily, be acute. If,
therefore, we take the formula for sin a, or for sin )8, according as the
force P, or the force Q, is the greater, we do away with all ambiguity,
and have merely to take the value of the angle shown in the table of
sines. And by subtracting the value thus found, from the given
value of t, we find the value, whether acute or obtuse, of the other of
the two angles, a, ^.
421. To determine a and /8 otherwise. After the magnitude of the
resultant has been found, we know the three sides, ]\IA^ AD^ MD^
of the triangle DMA, then we have
^,,, MD' + MA'-AD'
(4)
(5)
by successive applications of the elementary trigonometrical formula
used above for finding MD. Again, using this last-mentioned for-
mula for AID' or R^ in the numerators of (4) and (5), and reducing,
we have cosa = ^ -, (o)
r, 0 + RcosL . .
co5/3 = ^^-^ ; (7)
formulae which are convenient in many cases. There is no am-
biguity in the determination of either a or ;8 by any of the four
equations (4), (5), (6), (7).
Remark. — Either sign (4- or -) might be given to the radical
in (i), and the true line of action and the direction of the force in it
would be determined without ambiguity by substituting in (2) and
(3) the value of R with either sign prefixed. Since, however, there
can be no doubt as to the direction of the force indicated, it will be
generally convenient to give the positive sign to the value of R. But
in special cases, the negative sign, which with the proper interpre-
tation of the formulae will lead to the same result as the positive,
will be employed.
422. Another method of treating the general problem, which is
useful in many cases, is this :
Let
which implies that
P=F^G,
Q^F^G.
STATICS OF A PARTICLE. -^ATTRACTION. 145
^and G will be both positive if F> Q. Hence, instead of the two
given forces, F and Q. we may suppose that we have on the point M
four forces; — two, each equal to F, acting in
the same directions, MK, ML, as the given
forces, and two others, each equal to G, of
which one acts in the same direction, MK, as
F, and the other in ML\ the direction opposite
to Q. Now the resultant of the two equal
forces, F, bisects the angle between them,
KML) and by the investigation of § 423
below, its magnitude is found to be 2FC0S J t.
Again, the resultant of the two equal forces,
G, is similarly seen to bisect the angle, KMF ,
between the line of the given force, F, and
the continuation through M of the line of the given force, Q; and to
be equal to 2 6^ sin \ i, since the angle KLM' is the supplement of i.
Thus, instead of the two given forces in lines inclined to one another
at the angle t, which may be either an acute, an obtuse, or a right
angle, we have two forces, 2^cos \ i and 2 6^ sin J t, acting in lines,
MS, MT, which bisect the angles LMK and KML!, and therefore
are at right angles to one another. Now, according to § 429 below,
we find the resultant ^ of these two forces by means of the following
formulae : —
tan SMD = —= ^ ,
2I1 cos \ I
and R = 2jFcos \ i sec SMD,
F— 0
or tan(it-a)=-^-j-->tanJt, <8)
and R = {F^ Q)co?>\i sec (J t - a)
= (/'+0cosi(a + ^)cosi(a-^). (9)
These formulae might have been derived from the standard formulae
for the solution of a plane triangle when two sides {P and 0, and
the contained angle (tt - i) are given.
423. We shall now investigate some cases of the general formulae.
Case I. Let the forces be equal, that is, let Q. = F in the preceding
formulae.
Then, by (i), R^ ^ 2F' + 2F' cos i=2F'{i + cos i)
= 4/''cos'Jt.
Hence i?=2/'cosJt,
an important result which might, of course, have been obtained
directly from the proper geometrical construction in this case. Also
by (2),
1 In the diagram the direction of the balancing force is shown by the arrow-
head in the line DM.
T. 10
146 ABSTRACT DYNAMICS.
^ sin I C^ sin t . ,
sin a = — -— = -^ — = sin \ t,
which agrees with what we see intuitively, that a = /5 = J i.
424. Case II. Let F= Q ; and let t = 120^ Then
cos J t = cos 60° = I, and (§ 423) i? = P.
The resultant, therefore, of two equal forces inclined at an angle
of 120" is equal to each of them. This result is interesting, because
it can be obtained very simply, and quite independently of this
investigation. A consideration of the symmetry of the circumstances
will show that if three equal forces in one plane be applied to a
material point in lines dividing the space around it into three equal
angles, they must be in equilibrium; which is perfectly equivalent to
the preceding conclusion.
425. Case III. Let t = 0% cos t = i ;
then R = {I^+Q' + 2FQf,
R = F-^Q.
426. Case IV. Let t = 180"; cos t = - i ;
then R = {F' + Q'-2FQ)K
F = F-Q.
This is also one of the cases in which it is convenient to give some-
times the negative sign, sometimes the positive to the expression for
the resultant force : for if Q be greater than F, the preceding expression
will be negative, and the interpretation will be found by considering
that the force which vanishes when F= Q, is in the direction of F
when F is the greater, and in the contrary direction, or in that of Qy
when F is the less of the two forces.
427. Case F. Forces nearly conspiring. Let the angle t be very
small, then sintwt;^ cosi«i.
The general expressions {§419) therefore become,
R^F+Q,
Qi
sm^«^p^^.
a«:-
To the same degree of approximation
p^
Hence „ + ^«^. «i^^ =..
* The sign «s is used to denote approximate equality.
(.0)
STATICS OF A PARTICLE.— ATTRACTION. 147
This shows that the errors in the values of a and P obtained ap-
proximately by this method compensate; one being as much above,
as the other is below, the true value.
We therefore conclude that the resultant of two forces very nearly
conspiring is approximately equal to their sum, and approximately
divides the angle between them into parts inversely as the forces.
When the angle between the forces is infinitely small, they may
either conspire in acting on one point in one line ; or they may act
on different points in parallel Hues. In either case the resultant is
precisely equal to their sum. Actually conspiring forces we have
already considered ; parallel forces we shall consider more particularly
when we treat of the equilibrium of a rigid body. We may briefly
examine the case here however. Suppose the actual points of appli-
cation of the forces to be ^ . .
and B, but let their lines ^^__^_,-^
meet in a point M; join ^ " ^
AB, and let MAB be an M ■''^^s^i:!!^:::^^^^^-^^
isosceles triangle. Let this
point M be removed grad-
ually to an infinite distance in the direction of a perpendicular, OMy
bisecting the line AB. The resultant will still divide the angle in-
versely as the forces : and as the circular measure of the angle is any
arc described from M as centre divided by the radius, every such arc
will be divided in the same proportion. Now, if M be infinitely
distant, that is if the lines of the forces be parallel, the arc will become
a straight line, and will be divided into parts inversely as the forces.
In actual cases of forces acting on a point, and very nearly con-
spiring, the following approximate equations show how nearly the
resultant approaches the sum of the forces : —
smO^O] cos6'«i-i^'.
BT
Ji^(P^Q)-l^Qc\ (12)
that is, the resultant of two forces very nearly conspiring falls short of
their sum by the square of the angle between them multiplied into a
quarter of their harmonic mean^
428. Case VI. Forces nearly opposed.
1*'. Let the angle t be very obtuse, and the two forces exactly equal.
1 The Harmonic Mean of two numbers is the reciprocal of the mean of their
2PQ
reciprocals. Thus the harmonic mean of P and Q is „ „ .
10 — 2
148 ABSTRACT DYNAMICS.
Let t = TT - ^, where 6 is very small,
then J ' = i 7^ ~ i ^>
cos J t = sin I ^,
7?=2/'sin|^,
and since the sine of a very small angle is equal to the angle, in
circular measure R-^PB.
Hence the resultant of two equal very nearly opposed forces is
proportional to the defalcation from direct opposition: being ap-
proximately equal to either of the forces multiplied into the supple-
ment of the angle between them.
2°. If the forces are neither equal nor nearly equal, the resultant
will be approximately equal to their difference.
We have as before,
cos t«- I,
R^^F'+Q'-2FQ.
Therefore R^P-Q,
The ambiguity as to whether the acute angle, shown in the table,
or its supplement, is to be chosen in either case, may be removed by
considering which of the two forces is the greater.
Thus, as we suppose P to be greater than Q^ a is acute, and there-
fore sma«aw-!^:^ — -— ^
and p is obtuse.
Therefore ^8 « tt - ^j^'J^
r. Pl-Qit
P^-PZ^'
We find, by addition,
a + /3 = -^--^t = i,
and conclude, as in the former case, that the errors in the approximate
values of a and p compensate, one being as much above, as the other
is below, the true value.
It is only when R is comparable in magnitude with P and Q, that
the foregoing solution is applicable.
But if P exceeds Q, or if Q exceeds P, by any difference which is
considerable in comparison with either, the formulae hold.
Let us suppose now that, while P remains of any constant mag-
nitude, Q is made to increase from nothing, gradually, until it becomes
STATICS OF A PARTICLE.— ATTRACTION. 149
first equal to, and then greater than P, the angle t remaining constant.
The angle a will increase very slowly, according to the approximate
formula (10), until Q becomes nearly equal to P. Then as the value
of Q is increased until it becomes greater than P, the value of a will
•increase very rapidly through nearly two right angles, until it falls
but little short of t, when its supplement will be approximately ex-
pressed by the formula (10).
In this transition, from Q<P to Q>P, the direction and magni-
tude of the resultant are most conveniently found by means of (§ 422)
the last of the three general methods given above for determining the
resultant of two forces.
Thus, instead of the two given forces we may substitute two
forces in lines bisecting respectively the obtuse angle LMK, or t,
and the acute angle KML! and of magnitudes which approximate
to \(^P-\-Q) (tt-i), and P-Q, respectively, when t is nearly two
right angles.
We infer, finally, that, however nearly P and Q are equal to one
another, the approximate formulae of § 428, 2° hold, provided only
\ {P+ Q){tt- i) is a small fraction oi P~ Q.
429. Case VII. Let t = 90"; cos 1 = 0, sin t = 1 ;
then R = (P'+Q')K (13)
and sin a = ^
J I. 04)
sm/3 = -^
In this case, p being the complement of a, sin p = cos a.
Hence cos « = "d •
_ , . sin a
Lastly, since tan a = ,
•' cosa
we deduce tan a = -p, (15)
and R = Pseca. (16)
Remark. — These formulae have thus been derived from the general
expression (§ 419); but they can also be very readily got from a
special geometrical construction, corresponding to the case in which
the lines of the forces are at right angles to one another, the prin-
ciples to be used being (i) the parallelogram of forces; (2) Euchd L,
XLVIL; and (3) the trigonometrical definitions of sine, cosine, and
tangent.
430. This case is of importance, for It affords us the formulae for
rectangular resolution ; by the aid of which we shall, a little later,
proceed to calculate the resultant of any number of forces in one
plane. We might calculate the resultant by applying the elementary
15° ABSTRACT DYNAMICS.
formulae (§§ 419, 420, 421) to repetitions of the parallelogram of forces.
But this process would be very complicated and tedious, if the forces
were numerous, and their magnitudes and angles given in numbers;
and we shall see that it may be avoided by resolving all the forces
along two lines at right angles to one another, and thus obtaining
as equivalent to them, two forces along these lines.
We shall first consider the general inverse problem (§ 418), or the
resolution of forces.
431. If a force acting on a material point, and two lines in one
plane with the line of that force, be given, it is possible to find deter-
minately two forces along those lines, of which the given force is the
resultant.
The two forces thus determined are called the components of the
given force along the given lines, and if we substitute these two forces
for the given force, we are said to resolve the given force into two
forces along the given lines; or, to resolve the force along the given
lines.
Geometrical Solution. Let M be the given point ; 7?, the
given force acting on it in the line, MK',
and J/T^and MG the given fines.
It is required to find the components
along J/T^and MG of R in MK.
Take any convenient length MD to
represent the magnitude of the given
force, R. Through D draw DA parallel
to GM, and let it cut MF in A ; and
also through D draw DB parallel to FM^ and let it cut MG in B ;
MA and MB represent the required magnitudes of the components.
433. Trigonometrical Solution. If the angle KMF be given = a,
and KMG = /?, and if the required component of the given force R
along MF be denoted by P^ and the component along MG by Q, we
deduce from equations (2) and (3) (§ 420), the following : —
^~sin(a + ^)' ^'^^
^-sin(a + )8)- ('^>
434. When the given lines of resolution are at right angles to one
another, these expressions are modified in the manner shown above
(§ 429, Case VII), or we may find them
at once from the geometrical construc-
tion proper for the case, thus : —
Let MX, MY be the given lines;
XMY = 9o^ and MD = R. Also, as be-
fore, DMA = a, and DMB = jS. Draw
DA parallel to YM, or perpendicular to
MX, and make MB = AD. Then in the
STA TICS OF A PAR TICLE.-^A TTRA CTION. 1 5 1
right-angled triangle MAD, MA =M£> co^ DMA, and AD^MD
sin DMA.
Hence, since MA represents the component along MX, and MB
the component along MY,
F=Rcosa, (19)
(2 = i? sin a, or (2 = i? cos /?. (20)
Hence, in rectangular resolution, the component, along any line,
of a given force, is equal to the product of the number expressing the
given force, into the cosine of the angle at which its direction is in-
clined to that line.
435. Application of the Resolution of Forces.
number of forces jP^, P^, P^,F^, P^,
acting respectively in lines ML^^
ML,^, ML^, ML^, ML^, on a ma-
terial point M\ required their re-
sultant.
Through M, draw at right angles
to each other, and in the same plane
as the given forces, two lines, XX'
and YY', which may be called lines
or axes of resolution. Let the angle
which the resultant forms with the
line of resolution MX, be denoted
by 0, and let the angles, which the
lines of the forces make respectively
with the lines of resolution, be denoted by a^, ^^) a^, (3^; a^, p^; &c.;
that is, L^MX=a^, L^MY=^p^, and so on.
The angles /3j p„, &c., are merely the complements of a^ a^, &c.,
and, except for the"" sake of symmetry, they need not have been intro-
duced into our notation.
Resolve (§ 434) the first force P^, into two components, one along
MX, and one along MY. These are
P^ cos ttj along MX, which force may be denoted by X^,
and P^ sin a^ along MY, which force may be denoted by Y^.
Treat all the other forces in like manner, thus reducing them to
components along MX and MY] and add together the components
along each of the lines of resolution. Then if X denote the sum of
the components along MX, and y the sum of the components along
MY, we have
X=P^ cos a, + P^ cos a^ + P^ cos a^ + P^ COS a^ + P^ COS a^,
F= P^ sin ttj + P"^ sin a^ + P^ sin a^ + P^ sin a^+ P^ sin a^.
Lastly, to find the resultant of X and K
(§429). i?=v'(jr'+n
and cos ^--bj
(2,)
(22)
152 ABSTRACT DYNAMICS.
or, as is in general better for calculation,
Y
tan^ = -, (23)
whence we derive the magnitude of the resultant,
Ji = XsQce. (24)
The calculation will in general be facilitated by the use of log-'
arithms; for which purpose equations (25) and (24) are to be modified
in the following manner : —
tab. log. tan 6 = log. Y- log. X+ 10, (25)
log. R = log. X+ tab. log. sec. ^ - 10. (26)
Remark i. — It is to be observed that the sums X of the different
components X^., X^^ &c., and Fof Y^, Y^, &c., are got by an algebraic
addition, whatever may be the algebraic signs of the several terms.
If the given forces act all round the point M, it will happen in the
resolution that the different components do not all act in the same
directions along XX' and YY\ It will be necessary, therefore, to
fix upon one direction as positive. Thus, if MX and MY be posi-
tive directions, MX\ MY' will be negative; and absolute values of
the components, which act from M to X', and from M to F',
must be subtracted from, instead of added to, those along MX
2.ndMY.
Remark 2. — In choosing the axes of resolution, it simplifies the
problem to fix on one of the lines which represent the forces, as one
of the axes, and a line perpendicular to it, as the other.
Let J/Zj, the line of the first force P^, be the axis JOT, and MY,
a line perpendicular to it, the other,
a^ in this case is nothing; and the angle F^ MP^ = a^.
Hence, if a, = o, the resolution of the first force is
'X^=P^COSa^=Pj.f
P^ sin ttj = o,
that is, P^ requires no resolution.
If two of the forces happen to be at right angles, it will be con-
venient to choose the axes along them, and then neither requires
resolution.
Actual cases may often be simplified by observing if any two of the
forces are opposite, in which case, one force, equal to the excess of
the greater above the less, and acting in the direction of the greater,
may be taken instead of them.
Remark 3. — When the direction of the resultant is known, and its
magnitude is required, it is most convenient to make it one of the
axes of resolution.
•'{%
STATICS OF A PARTICLE.— ATTRACTION. 153
Let MK be the direction of the
resultant of F^, F^, F^, F^, the dif-
ferent forces. Resolve each force
into two, one along MK, and one
in a hne perpendicular to it. Add
the components along MK. The
sum must be the magnitude of the
resultant; and the components along
the other line must balance one an-
other. Hence,
X=^R = F^ cos A, MKh F^ cos A^MK-v &c.,
and Y= F^ sin A^ MK+ F^ sin A^ MK+ &c. - o.
Remark 4. — Equations (23) and (24) may be employed with ad-
vantage in all cases where the numbers of significant figures in the
values to be used for X and Fare large.
By equations (23) and (24) the direction of the resultant is first
determined, and then its magnitude, not as in equations (21) and (22),
the magnitude first, and then the direction.
436. For the better understanding of what follows a slight digres-
sion (§§ 437, 464) upon projections and geometrical co-ordinates is
now inserted.
437. The projection of a point on a straight line, is the point in
which the latter is cut by a perpendicular to it from the former.
438. Any line, joining two points, is called an arc. It is not
necessary to confine this expression to its most usual signification of
a continuous curve line. It may be appfied to a straight line joining
two points, as an extreme case; or it may be applied to a zigzag or
angular path from one point to the other; or to a self-cutting path,
whether curved or polygonal; in short, to any track whatever, from
one point to the other.
♦
439. The projection of an arc on a straight line, is the portion of
the latter intercepted between the projections of the extremities of the
former.
440. If we imagine an arc divided into any number of parts, the
projections of these parts, taken consecutively on any straight line,
make up consecutively the projection of the whole. Hence, the sum
of the projections of the parts is equal to the projection of the whole.
But in this statement, it must be understood that, of such partial
projections laid down in order, those which are drawn in one di-
rection, or forwards, being reckoned as positive, those which are
drawn in the other direction, or backwards, must be reckoned as
negative.
441. The projection of an arc on any straight line, is equal to the
length of the straight line joining the extremities of the former, mul-
154* ABSTRACT DYNAMICS.
tiplied by the cosine of the angle* at which it is inclined to the latter.
This angle, if not a right angle, will be acute or obtuse, according to
the convention which is understood as to the direction reckoned
positive in the line of projection ; and the extremity of the arc which
is Xd^tn first in drawing 2, positive line from one extremity of it to the
other.
442. The orthogonal projection of a line, straight or curved,
closed or not closed, on a plane, is the locus of the points in which
the latter is cut by perpendiculars to it from all points of the former.
Other kinds of projections are also used in geometry; but when no
other designation is applied or understood, the simple ttrm projectio?i
will always mean orthogonal projection.
443. A circuit is a line returning into itself, or a line without ends
in a finite space. It is (if a continuous curve) often called a closed
curve; or if made up altogether of rectilinear parts, a closed polygoii.
A circuit in one plane may be either simple or self-cutting. The latter
variety has been called by De Morgan, autotomic. But whether simple
or autotomic, there is just one definite course to go round a circuit;
and at double or multiple points, this course must be distinctly
indicated^ (arrow-heads being generally used for the purpose on
a diagram, like the finger-posts where two or more roads cross). A
circuit not confined to one plane need never be considered to be
autotomic, unless as an extreme case. Thus, if we take any thread or
wire, however fine, and bend it into any curve or broken line, or tie it
into the most complicated knot or succession of knots, but attach its
ends together; any geometrical line drawn altogether within it, from
any one point of it, round through its length back to the same point,
constitutes essentially a simple or not self-cutting circuit.
444. 'The area enclosed by,' or *the area of a simple plane
circuit, is an expression which requires no explanation. But, as has
been shown by De Morgan^, a peculiar rule of interpretation is
necessary to apply the same expression to an autotomic plane circuit,
and it has no apphcation, hitherto defined, to a circuit not confined to
one plane.
445. The area of an autotomic plane circuit, is the sum of the
areas of all its parts each multiplied by zero with unity as many times
added as the circuit is crossed'* from right to left, and unity as many
1 The angle at which one line is inclined to another, is the angle between two
lines drawn parallel to them from any point, in directions similar to the directions
in the given lines which are reckoned positive.
" 'A curve which has double or multiple points, may be in many different
ways a circuit, or mode of proceeding from one point to the same again. Thus the
figure of 8 may be traced as a self -cutting circuit, in the way in which it is natural
if the curve be a continuous lemtiiscate, or it may be traced as a circuit presenting
two coincident salient points. A determinate area requires a determinate mode of
making the circuit.' De Morgan, Cambridge and Dublin Mathematical yournal^
May, 1850.
^ * Extension of the word area,' Cambridge and Dublin Mathematical yournal^
May, 1850.
^ A moving point is said to cross a plane circuit from right to left, if it crosses
STATICS OF A PARTICLE.— ATTRACTION. 155
times subtracted as the circuit is crossed from left to right, when a
point is carried in the plane from the outside to any position within
the enclosed area in question. The diagram, which is that given by De
Morgan, will show more clearly what is meant by this use of the word
area. The reader, with this as a model, may exercise himself by
drawing autotomic circuits and numbering the different portions of
the enclosed area according to the rule, which he will then find no
difficulty in understanding.
446. Any portion of surface, edged or bounded by a circuit, is
called a skeiL
A plane area may be regarded as an extreme case, but generally
the surface of a shell will be supposed to be curved.
A simple shell is a shell of which the surface is single throughout.
One side of the shell must always be distinguished from the other,
whatever may be the convolutions of its surface. Thus we shall have
a marked and unmarked side, or an outside and an inside, to dis-
tinguish from one another.
447. The projection of a shell on any plane, is the area included
in the projection of its bounding line.
448. If we imagine a shell divided into any number of parts, the
projections of these parts on any plane make up the projection of the
whole. But in this statement it must be understood that the areas of
partial projections are to be reckoned as positive only if the marked
side, or, as we shall call it, the outside, of the projected area, and a
marked side, which we shall call the front, of the plane of projection,
face the same way.
If the outside of any portion of the projected area faces on the
whole backwards, relatively to the front of the plane of projection, the
projection of this portion is to be reckoned as negative in the sum.
from the right side to the left side as regarded by a person looking from any point
of the circuit in the direction reckoned positive.
1 56 ABSTRA CT D YNAMICS.
Of course if the projected surface, or any part of it, be a plane area
at right angles to the plane of projection, the projection vanishes.
Cor. The projections of any two shells having a common edge, on
any plane, are equal. The projection of a closed surface (or a shell
with evanescent edge), on any plane, is nothing.
449. Equal areas in one plane or in parallel planes, have equal
projections on any plane, whatever may be their figures. [The proof
is easily found.]
Hence the projection of any plane figure, or of any shell, edged by
a plane figure, on another plane, is equal to its area, multiplied by the
cosine of the angle at which its plane is inclined to the plane of pro-
jection. This angle is acute or obtuse, according as the marked sides
of the projected area, and of the plane of projection face, on the whole,
towards the same parts, or on the whole oppositely.
450. Two rectangles, with a common edge, but not in one plane,
have their projection on any other plane, equal to that of one rect-
angle, having their two remote sides for one pair of its opposite sides.
For, the sides of this last-mentioned rectangle constitute the edge of
a shelly which we may make by applying two equal and parallel
triangular areas to the sides of the given rectangles; and the sum of
the projections of these two triangles on any plane, according to the
rule of § 448, is nothing.
Hence (as is shown by a very simple geometrical proof, which is
left as an exercise to the student), we have the following construction
to find a single plane area whose projection on any plane is equal to
the sum of the projections of any two given plane areas.
From any convenient point of reference draw straight lines per-
pendicular to the two given plane zx^z.^ forward^ relatively to their
marked sides considered as fronts. Make these lines numerically
equal to the two areas respectively. On these describe a parallel-
ogram, and draw the diagonal of this parallelogram through the point
of reference. Place an area with one side marked as front, in any
position perpendicular to this diagonal, facing forwards, and relatively
to the direction in which it is drawn from the point of reference.
Make this area equal numerically to the diagonal. Its projection on
any plane will be equal to the sum of the projections of the two given
areas, on the same plane.
The same construction maybe continued; just as, in § 413, the
geometrical construction to find the resultant of any number of
forces; and thus we find a single plane area whose projection on any
plane is equal to the sum of the projections on the same plane of any
given plane areas. And as any shell may (if it be not composed of a
finite) be regarded as composed of an infinite number of plane areas,
the same construction is applicable to a shell. Hence the projection
of a shell on any plane is equal to the projection on the same plane,
of a certain plane area, determined by the preceding construction.
From this it appears that the projection of a shell is nothing on
STATICS OF A PARTICLE.—ATTRACTION. 157
any plane perpendicular to the one plane on which its projection
is greater than on any other; and that the projection on any inter-
mediate plane is equal to the greatest projection multiplied by the
cosine of the inclination of the plane of the supposed projection to
the plane of greatest projection.
451. To specify a point is to state precisely its position. As we
have no conception of position, except in so far as it is relative, the
specification of a point requires definite objects of reference, that is,
objects to which it may be referred. The means employed for this
purpose are certain elements called co-ordinates, from the system of
specification which Descartes first introduced into mathematics. This
system seems to have originated in the following method, for de-
scribing a curve by a table of numbers, or by an equation.
452. Given a plane curve, a fixed line in its plane, and a fixed
point in this line, choose as many points in the curve as are required
to indicate sufficiently its form: draw perpendiculars from them to
the fixed line, and measure the distances along it, cut oft' by these
lines, reckoning from the fixed point. In this way any number of
points in the curve were specified. The parts thus cut off along the
fixed line, were termed li7ieae abscissae^ and the perpendiculars, lineae
ordinaiim applicatae. The system was afterwards improved by draw-
ing through the point of reference a line at right angles to the first,
and measuring off along it the ot'dmafes of the curve. The two lines
at right angles to one another are called the axes of reference, or the
lines of reference. The ordinate and abscissa of any point are termed
its co-ordinates; and an equation between them, by which either may
be calculated when the other is given, expresses the curve in a per-
fectly full and precise manner.
453. It is not necessary that the lines of reference be chosen at
right angles to each other. But when they are chosen, inclined at
any other angle than a right angle, the co-ordinates of the point
specified are not its perpendicular distances from them, but its
distances from either, measured parallel to the other. Such oblique
co-ordinates are sometimes convenient, but rectangular co-ordinates
are, in general, the most useful; these we shall now consider.
454. If the points to be specified are all in one plane, the objects
of reference are two lines at right angles to
one another in that plane. Thus, let P
be a point in a plane XOY; and let OX,
OY,he two lines in the plane, cutting each
other at right angles in the point O. Then
will the position of the point F be known,
if the perpendicular distance of the point
F from the line OX, namely, the length
of the line FA, and the perpendicular dis-
tance from OY, namely, the length of the
Hne FB, be known.
158
ABSTRACT DYNAMICS.
455. Again, let points, not all in one
plane, but in any positions through space
be considered. To specify each point now,
three co-ordinates are required, and the
objects of reference chosen may be three
planes at right angles to one another; thus,
the point P is specified by the lines FK^
FH, FI, drawn perpendicular to the planes
'^ YZ, ZX, XY, respectively.
In our standard diagrams the positive
directions OX, O Y, OZ, are so taken that
if a watch is held in the plane XO Y, with its face towards OZ, an
angular motion against the hands would carry a line from OX to O K,
through the right angle XO Y.
456. When the objects to be specified are Hnes all passing through
one point, the specifying elements employed, are angles standing in
definite relation to them, and to the objects of reference. There are
two chief modes in which this kind of specification is carried out :
the polar and the symmetrical.
457. Fo/ar Method. 1°. Lines all in one plane. In this case the
object of reference is any fixed line through their common point of
intersection, and lying in their plane.
Let O be the common point of intersection,
^^^__^^^ OX the fixed line, and OF the line to be
0^^""^^ X specified. Then the position of OF will be
known, if the angle XOF^ which the line OF
makes with OX, be known.
2°. Lines in space, all passing through one point, may be specified
by reference to a plane and a line in it, both passing through their
common point of intersection.
Let OF be one of a number of lines, all
passing through O, to be specified with refer-
ence to the plane XO K, and the line OX in it.
Through OF let a plane be drawn, cutting the
plane XO Y at right angles in OE. Then the
line OF will be specified, if the angles XOE,
EOF are given.
Corollary. Similarly, if the line OF be the
locus of a series of points, any one of these
points will be specified, if its distance from O
and the two angles specifying the line OFj are
known.
458. Symmetrical Method. In this method the objects of reference
are three lines at right angles to each other, through the common
point of intersection of the lines to be specified, and the specifying
elements are the three angles which each line makes with these three
lines of reference.
Thus, if O be the common point of intersection, OK one of the
STATICS OF A PARTICLE.— ATTRACTION. 159
lines to be specified, and OX, OV, OZ, the lines of reference; then
the angles XOK, YOK, ZOK, are the specifying elements.
459. From what has now been said, it will be seen that the pro-
jections of a given line on other three at right angles to each other
are immediately expressible, if its direction is specified by either of
the two methods.
1°. Fo/ar Method. Let OK be the given line,
and OX, OV, OZ, the lines along which it is to be
projected. Through OZ and OX let a plane pass,
cutting the plane XOV in OF. Through X draw
another plane, KEA, cutting OX perpendicularly in
A and KEB cutting O Y perpendicularly in B.
Then KE, being the intersection of two planes
each perpendicular to XOY, is perpendicular to
every line in this plane. Hence, OEK is a right
angle.
Hence,
OE^OKq.q^KOE.
Again, since the plsine XAE was drawn perpendicular to OX,
OAE is a right angle.
Hence, OA = OF cos FOX= OX cos KOF cos EOX,
and similarly, OB = OF cos FO Y= OX cos XOF cos EO Y,
or if we put
OX=r, XOF = <i>, FOK=i, KOZ=Q = \Tr-i,
and let the required lines be denoted by x, y, z, then
^ = r sin ^ cos <^,)
y = rsm 6s\n ^,> (i)
z = r cos 6. )
2°. Synwtetrical Method. Let the line be referred to rectangular
axes by the three angles, a = ^(9X, 13 = XOY, y = XOZ.
Then the required projections are
^ = ^cosa, y = r cos P, z = r cosy.
460. Referring again to the diagram, we have
OF'=OA'+OB',
and OX' = 'OF'+OC',
therefore, OX' = OA' + OB' + 0C\
or r' = x' -^y' + z'. (2)
Substituting here for x, y, z, their values, in terms of r, a, p, y,
found above, and dividing both members of the resulting equation by
f-^, we have
I = cos^ a ■\- cos^ P + cos^ y. (3)
461. In the symmetrical method, three angles are used; but, as we
have seen, only two are necessary to fix the position of the line. We
i6o ABSTRACT DYNAMICS,
now see that, if two of the three angles, a, ft y, are given, the third
can be found. Suppose a and ^ given, then by § 460,
cos^ y = I - cos* a - cos^ ft
For logarithmic calculation, the following modification of the pre-
ceding formula is useful,
cos^ y = sin^ a - cos^ i^ = - cos (a + ft x cos (a - ft,
whence cos y = ^{- cos (a + ft x cos (« - ft}
= \/{cos (tt - a - ft X cos (a - /8)},
Tab. Log. cos y = 1 {T. L. cos (tt - a - ^) + T. L. cos (a - ft}. (4)
462. The following comparison will show in what way the two
systems are related, and how it is possible to derive the specifying
elements of either from those of the other. In the polar method, the
■ fixed line in the equatorial plane, corresponds with one of the three
lines of reference in the symmetrical. A line in the equatorial plane,
drawn at right angles to the fixed line of the polar system, constitutes
a second line of reference in the symmetrical system. The third line
in the symmetrical system, is the axis of the polar system, from
which the polar distance \B) is measured. A comparison of pre-
ceding formulae shows that .
cos a = sin ^ cos ^,)
cos y8 = sin ^ sin <^, > . (5)
cos y = cos B, )
463. The cosines of the three angles, a, ft y, of the symmetrical
system, are commonly called the direction cosines of the line specified.
If we denote them by /, w, n^ we have as above,
P^■m^ + n^=\. (6)
A line thus specified is for brevity called the line (/, ;;?, n).
\il^ 7n, ?t, are the direction cosines of a certain line; it is clear that
- /, - m, — ;/, are the direction cosines of the line in the opposite
direction from O. Thus it appears that the direction cosines of the
line, specify not only the straight line in which it lies, but the direction
in it which is reckoned as positive.
464. We conclude this digression with some applications of the
principles explained in it, which are useful in many dynamical in-
vestigations.
{a) To find the mutual inclination, 6, of two lines, (/, ;«, «),
(/, 7n', n'). Measure off any length 0K= r, along the first line (see
fig. of § 459). We have, as above,
OA=lr, AE = mr, EK=nr.
Now (§ 441), the projection of OK on the second line, is equal to
the sum of the projections of OA^ AE, EK, on the same. But the
cosines of the angles at which these several lengths are inclined to the
line of projection, are respectively cos ^, /, m', n'. Hence
OK cos 6 = OAf + AEm' + EKn'.
STATICS OF A PARTICLE.— ATTRACTION. i6i
If we substitute in this, for OK, OA, AE, EK, their values shown
above, and divide both members by r, it becomes
cos 9 = 11' + mm' + ;?;/, (7)
a most important and useful formula.
Sometimes it is useful to have the sine instead of the cosine of 0.
To find it we have of course,
sin^ ^ = I - (// + mm! + ni^Y.
This expression may be modified thus: — instead of i, take what is
equal to it, {P + m'^ + //'') (/' + m'-\- tf),
and the second member of the preceding becomes
(P + m' + n') {P + 7n'' + n"') - (// + mm' + nn'Y
= {mji'Y + {nm'Y - 2mm' nn' + &c. = {mn' - nm'Y + &c.
Hence, sin 0 = {{mn' - nm'Y + (nr - /n'Y + {/m' - mfY}^, (8)
(d) To find the direction cosines. A, fj., v, of the common perpen-
dicular to two lines, (/, pi, n), (/, m', n').
The cosine of the inclination of (X, /j., v) to (/, m, n) is, according
to (7) above, /X + mfx + nv, and therefore
/X + mix + nv = o,)
similarly A + m'fi + nv =o,y (9)
also (§463) X' + tJi' + v' = i.)
These three equations suffice to determine the three unknown
quantities. A, fx, v. Thus, from the first two of them, we have
X _ /A _ V , >
mn' — nm' nl' — In' Im' - I'm ' ^
From these and the third of (9), we conclude
^ mn' - nm' „
{{mn'-nm'Y^ Inl'-ln'Y ■¥ {Im' -m^Y)^
or if we denote, as above, by 6, the mutual inclination of (/, m, n)
. {mn' — nm') {nV — hi) {Int — ml') . .
A= -. J , IK — : -^ , V= ^ J. . (11)
sm 6> ^ sm ^ ' sm ^ ^
The sign of each of these three expressions may be changed, in as
much as either sign may be given to the numerical value found for
sin 0 by (8). But as they stand, if sin 0 is taken positive, they express
the direction cosines of the perpendicular drawn from O through the
face of a watch, held in the plane (/, m, n), {/', m', n'), and so facing
that angular motion, against or with the hands, would carry a line
from the direction, (/, m, 71), through an angle less than 180° to the
direction, (/', m', n'), according as angular motion, through a right
angle from OX to (9 F is against or ivith the ha?ids of a watch, held in
T. II
1 62 ABSTRACT DYNAMICS.
the plane XOY, and facing towards OZ. This rule is proved by
supposing, as a particular case, the lines (/, ;;z, n), (/', m\ n'), to
coincide with OX and (9 F respectively; and then supposing them
altered in their mutual inclination to any other angle between o and
TT, and their plane turned to any position whatever.
If we measure off any lengths, OK-^r, and OK' = r'^ along the
two lines, (/, m^ n) and (/', 7n', n'\ and describe a parallelogram
upon them, its area is equal to r/ sin 6^ since r' sin 6 is the length of
the perpendicular from K' to OK. Hence, using the preceding
expression (8) for sin 6, and taking
Ir = X, mr = j, nr = 0,
// = x\ m'r' = y , « V = z\
we conclude the following propositions.
if) The area of a parallelogram described upon lines from the
origin of co-ordinates to points (^, 7, 0), (x', y^ z') is equal to
{{y^ -y'^y + {^^' - ^'^y + {^y - ^'yy^- (^ 2)
And, as X, ja, v, are the cosines of the angles at which the plane of
this area is inclined to the planes of YZj ZX, XY, respectively, its
projections on these planes are
yz' -y'z, zx' -z'x, xy'-xy. (13)
The figures of these projections are parallelograms in the three
planes of reference; that in the plane YZ, for instance, being de-
scribed on lines drawn from the origin to the points {y, z) and {y', z).
It is easy to prove this (and, of course, the corresponding expressions
for the two other planes of reference,) by elementary geometry.
Thus, it is easy to obtain a simple geometrical demonstration of the
equations (8) and (11). It is sufficient here to suggest this investi-
gation as an exercise to the student. It essentially and obviously
includes the rule of signs] stated above (§ 464 {a)).
{d) The volume of a parallelepiped described on OK^ OK, 0K\
three lines drawn from O to three points
{x, y, z), {x\ y\ z\ (x'\ y'\ z'% is equal to
x" {yz' -y'z) +y" {zx' - z'x) + z" (x/ - x'y), (14)
an expression which is essentially positive, if OK, OK, OK", are
arranged in order similarly to OX, OY, OZ {see § 455 above). The
proof is left as an exercise for the student.
In modern algebra, this expression is called a determinant, and is
written thus : —
X, y, z,
oo\ y, z', (15)
465. To find the resultant of three forces acting on a material
point in lines at right angles to one another.
^^^
X-
STATICS OF A PARTICLE.— ATTRACTION. 163
1° To find the magnitude of the resultant. F j^
Let the forces be given numerically, X, Y, Z,
and let them be represented respectively by
the lines MA, MB, MC at right angles to
one another.
First determine the resultant of X and Y
in magnitude. If we denote it by R\ we
have (§ 429)
R'=j{X' + Y'). (i) ^r A
This resultant, represented by ME, lies in the plane BMA\
and since the Hues of the forces X and Y are perpendicular to the
Hne MC, the Hne ME must also be perpendicular to it; for, if a line
be perpetidicular to two other lines, it is perpendicular to every other line
in their plane; hence R' acts perpendicularly to Z.
Next, find the resultant of R' and Z, the third force. If we denote
it by R, we have
R=J{R''+Z'),
and substituting for R'^ its value, we have
R=J{X' + Y' + Z'). (2)
2°. To find the direction of the resultant. Determine first the
inclination of the subsidiary resultant R' to MA or MB. Let the
angle EMA be denoted by <^ ; then we have
Next, let 7 denote the angle at which the line MZ> is inclined to
MC; that is, the angle CMD\ we have
cos7 = -^. (3)
Thus, by means of the two angles y and <^, the position of the line
MD, and, consequently, that of the resultant is found.
466. In the numerical solution of actual cases, it will generally be
found most convenient to calculate the three elements in the following
order: 1°, </>, 2°, y, 3°, R.
1°. To calculate ^, the formula already given, may be taken
Y
Un<j> = ^. (4)
(5)
(6)
(7)
II — 2
2°.
To calculate y.
We have
tany = -^.
But
R' = Xsec<f>.
Hence
X sec <j>
3^^.
To calculate R.
R = Zsecy.
i64 ABSTRACT DYNAMICS.
467. The angles determined by these equations specify the line of
the resultant, by what was called in previous sections (^ 457, 459)
the Polar Method.
The symmetrical specification of the resultant is to be found thus :
Let (in fig. of § 465) the angles at which the line of the resultant,
MD^ is incUned to those of the forces be respectively denoted by a, ^,
and y. Then, as above (equation (3)),
Z
cosy=-^. . (8)
By the same method we shall find
cos a = - , (9)
Y
and cos/8^^. (10)
If, therefore, there are three forces at right angles to one another,
the cosine of the inclination of their resultant to any one of them is
equal to- this force divided by the resultant.
This method requires that the magnitude of the resultant be known
before its position is determined. For the latter purpose, any two of
the angles, as was shown in Chapter V, are sufficient.
468. We shall now consider the resolution of forces along three
specified lines. The most important case of all is that in which the
lines are at right angles to one another.
Let the force R^ given to be resolved, be represented by MD^
and let the angles which it forms with the lines of resolution be
given, either a, ^, y, or y, <^. Required the components X^ K, Z.
1°. Suppose a, ^, y are given, then we deduce
from equation (9) X-Rq,o^ a;
from equation (10) Y=R cos p ;
and from equation (8) Z=R cos y.
2°. Suppose the data are R, y, <^, that is, the magnitude of the
resultant, its inclination to one of the axes of resolution, and the
inclination of the plane of the resultant and that axis to either of the
other axes.
To find the components X and V : resolve the force R in the
vertical plane CMED into two rectangular components along MC
and ME. Let the angle CMD be denoted by y. Then we have
for the component along MC^
Z=R cosy^ (11)
and for the component along ME,
ME = R sin y.
Next, resolve the component along ME in the horizontal plane
BMAEy into two, one along MAy and the other along MB. Let
STATICS OF A PARTICLE.— ATTRACTION, 165
Hie angle EMA be denoted by <^. Then we have for the com-
ponent along MA,
X =- ME cos cl> = R sin y cos 4>, (12)
and for the component along MB,
V= ME sin <\i = R sm y sin <^. (13)
469. We are now prepared to solve the general problem : — Given,
any number of forces acting on one point, in lines which lie in
different planes, required their resultant in position and magni-
tude.
Through the point acted on, draw three lines or axes of resolution
at right angles to one another. Resolve each force, by § 468, 1°, or
by § 468, 2°, into three components, acting respectively along the
three lines. When all the forces have been thus treated, add severally
the sets of components : by this means, all the forces are reduced to
three at right angles to one another. Find, by equation (2), their
resultant: the single force thus obtained is the resultant of the given
forces, which was to be found.
Remark. — All the remarks made with reference to the resolution
and composition of forces along two axes (§ 435) apply, with the ne-
cessary extension, to that of forces along three.
470. We are now prepared to answer the question which forms
the first general head of Statics ; What are the conditions of Equi-
librium of a material point 1 The answer may be put in one or other
of two forms.
1°. If a set of forces acting on a material point be in equilibrium,
any one of them must be equal and opposite to the resultant of the
others: or,
2°. If a set of forces acting on a material point be in equilibrium,
the resultant of the whole set must be equal to nothing.
471. Let us consider the first of these statements.
Given, a set of forces, F^, F^, F^, &c., in equilibrium : the force
/*,, for example, is equal and opposite to the resultant of F^, F.^, &c. ;
or, the resultant of F^, F^, &c., is —F^. Omitting F^, find the
resultant of the remaining forces by the general method ; the com-
ponents of this resultant will be
/*3 cos a^ + F3 cos a, + &c. along MX.
F^ cos p^ + F^ cos /?3 -t- &c. along MY.
F^ cos y^ + F^ cos yg + &c. along MZ.
Now, if — F^ be the resultant, the components of - F^ will be
equivalent respectively to the components of this resultant, there-
fore
- Pj cos a.^ = F^ cos a^ -I- F^ cos ttg + &c.
- /*j cos j8j = F^ cos ^2 + F.^ cos /?3 + &c.
- F^ cos y, -^- F^^ cos y, 4- P,^ COS y., + &c.
i66 ABSTRACT DYNAMICS.
Which equations, in the following more general form, express the
required conditions:
F^ cos a^ + P^ cos a^ + P.^ COS ttg + &c. = o.
P^ cos ^j + P^ cos /Sg + P^ cos ^3 + &c. = o.
P^ cos yj + /'g cos y^ 4- 7^3 cos y3 + &c. = o.
472. The second form of the answer may be illustrated either a^
dynamically, or b^ algebraically.
{a) Suppose all the forces reduced to three, X, F, Z, acting at right
angles to each other. Under what circumstances will three forces
give a vanishing resultant .? Substitute for X and Y their resultant
R\ and consider R' and Z at right angles to one another. If they
give a vanishing resultant, that is, if Z and R! balance, they must
either be equal and directly opposed, or else they must each be equal
to nothing. But they are not directly opposed, therefore each is
equal to nothing. Now, since R' = o, X and Y, v/hich are equi-
valent to R\ must also each be equal to nothing: in order, therefore,
that the resultant of forces acting along three lines at right angles to
one another may vanish, we have
P^ cos a^ + P^ COS a^ + &C. = o.
Pi COS ySi + P^ COS 1^1 + &C. = O.
P^ COS yi 4- P^ COS y^ + &c. = o.
{b) The general expression for the resultant is
R' = X' + Y' + Z\
Now, for equilibrium, R = o,
and therefore, X' + Y' + Z' = o.
But the sum of three positive quantities can be equal to nothing, only
when each of them is nothing : hence
X=o,
Y=o,
Z=o.
473. We may take one or two particular cases as examples of the
general results above. Thus,
1. If the particle rest on a smooth curve, the resolved force along
the curve must vanish.
2. If the curve be rough, the resultant force along it must be
balanced by the friction.
3. If the particle rest on a smooth surface, the resultant of the
applied forces must evidently be perpendicular to the surface.
4. If it rest on a rough surface, friction will be called into play,
resisting motion along the surface; and there will be equilibrium at
any point within a certain boundary, determined by the condition
that at z/ the friction is fx times the normal pressure on the surface,
while within it the friction bears a less ratio to the normal pressure.
When the only applied force is gravity, we have a very simple result,
which is often practically useful. Let 6 be the angle between the
STATICS OF A PARTICLE.— ATTRACTION. 167
normal to the surface and the vertical at any point; the normal
pressure on the surface is evidently W co^ 6^ where Wh the weight
of the particle ; and the resolved part of the weight parallel to the
surface, which must of course be balanced by the friction, is W sin 0.
In the limiting position, when sliding is just about to commence, the
greatest possible amount of statical friction is called into play, and
we have J^ sin 6 = ijlW cos $,
or tan 6 = 11.
The value of 6 thus found is called the Angle of Repose, and may
be seen in nature in the case of sand-heaps, and slopes formed by
debris from a disintegrating cliff (especially of a flat or laminated
character), on which the lines of greatest slope are inclined to the
horizon at an angle determined by this consideration.
474. A most important case of the composition of forces acting
at one point is furnished by the consideration of the attraction of a
body of any form upon a material particle anywhere situated. Experi-
ment has shown that the attraction exerted by any portion of matter
upon another is not modified by the neighbourhood, or even by the
interposition, of other matter; and thus the attraction of a body on a
particle is the resultant of the several attractions exerted by its parts.
To treatises on applied mathematics we must refer for the examina-
tion of the consequences, often very curious, of various laws of
attraction; but, dealing with Natural Philosophy, we confine our-
selves to the law of gravitation, which, indeed, furnishes us with an
ample supply of most interesting as well as useful results.
475. This law, which (as a property of matter) will be carefully
considered in the next Division of this Treatise, may be thus
enunciated.
Every particle of matter in the universe attracts every other particle
with a force, whose direction is that of the line joijiing the two, and
whose magnitude is directly as the product of their masses, and inversely
as the square of their distance from each other.
Experiment shows (as will be seen further on) that the same law
holds for electric and magnetic attractions ; and it is probable that it
is the fundamental law of all natural action, at least when the acting
bodies are not in actual contact.
476. For the special applications of Statical principles to which
we proceed, it will be convenient to use a special unit of mass, or
quantity of matter, and corresponding units for the measurement of
electricity and magnetism.
Thus if, in accordance with the physical law enunciated in § 475,
we take as the expression for the forces exerted on each other by
masses M and m^ at distance D, the quantity
Mm ^
it is obvious that our unit force is the mutual attraction of two units
of mass placed at unit of distance from each other.
1 6^ ABSTRA CT D YNAMICS.
477. It is convenient for many applications to speak of the density
of a distribution of matter, electricity, etc., along a line, over a sur-
face, or through a volume.
Here density of line is the quantity of matter per unit of length.
„ „ surface „ „ „ „ area.
„ ,, volume „ - „ „ „ volume.
478. In applying the succeeding investigations to electricity or
magnetism, it is only necessary to premise that M and m stand for
quantities of free electricity or magnetism, whatever these may be, and
that here the idea of mass as depending on inertia is not necessarily
Mm
involved. The formula -jr^ will still represent the mutual action, if
we take as unit of imaginary electric or magnetic matter, such a quan-
tity as exerts unit force on an equal quantity at unit distance. Here,
however, one or both of M, m may be negative ; and, as in these
applications like kinds repel each other, the mutual action will be
attraction or repulsion, according as its sign is negative or positive.
With these provisos, the following theory is applicable to any of the
above-mentioned classes of forces. We commence with a few simple
cases which can be completely treated by means of elementary geo-
metry.
479. If the different points of a spherical surface attract equally
with forces varying inversely as the squares of the distances, a particle
placed witJwi the surface is not attracted in any direction.
Let HIKL be the spherical surface, and P the particle within it.
Let two hues HK^ IL, intercepting very small arcs HI, KL, be
drawn through F; then, on account of the
similar triangles IIFI, KFL, those arcs will
be proportional to the distances HF, IF;
and any small elements of the spherical sur-
face atZT/and KI, each bounded all round
by straight lines passing through F [and very
nearly coinciding with IIK\ will be in the
duplicate ratio of those lines. Hence the
forces exercised by the matter of these ele-
ments on the particle F are equal ; for they
are as the quantities of matter directly, and the squares of the
distances, inversely; and these two ratios compounded give that of
equality. The attractions therefore, being equal and opposite, de-
stroy one another: and a similar proof shows that all the attractions
due to the whole spherical surface are destroyed by contrary attrac-
tions. Hence the particle Z' is not urged in any direction by these
attractions.
480. The division of a spherical surface into infinitely small ele-
ments, will frequently occur in the investigations which follow: and
Newton's method, described in the preceding demonstration, in which
the division is effected in such a manner that all the parts may be
taken together m pairs of opposite eletnents with reference to an internal
STATICS OF A PARTICLE.— ATTRACTION. 169
point ; besides other methods deduced from it, suitable to the special
problems to be examined ; will be repeatedly employed. The follow-
ing digression (§§ 481, 486), in which some definitions and elemen-
tary geometrical propositions regarding this subject are laid down,
will simplify the subsequent demonstrations, both by enabling us,
through the use of convenient terms, to avoid circumlocution, and
by affording us convenient means of reference for elementary prin-
ciples, regarding which repeated explanations might otherwise be
necessary.
481. If a straight line which constantly passes through a fixed
point be moved in any manner, it is said to describe, or generate,
a conical surface of which the fixed point is the vertex.
If the generating line be carried from a given position continuously
through any series of positions, no two of which coincide, till it is
brought back to the first, the entire line on the two sides of the fixed
point will generate a complete conical surface, consisting of two
sheets, which are called vertical or opposite cones. Thus the elements
ZT/and KL, described in Newton's demonstration given above, may
be considered as being cut from the spherical surface by two opposite
cones having P for their common vertex.
482. If any number of spheres be described from the vertex of a
cone as centre, the segments cut from the concentric spherical sur-
faces will be similar, and their areas will be as the squares of the
radii. The quotient obtained by dividing the area of one of these
segments by the square of the radius of the spherical surface from
which it is cut, is taken as the measure of the solid attgle of the cone.
The segments of the same spherical surfaces made by the opposite
cone, are respectively equal and similar to the former. Hence the
solid angles of two vertical or opposite cones are equal : either may
be taken as the solid angle of the complete conical surface, of which
the opposite cones are the two sheets.
483. Since the area of a spherical surface is equal to the square of
its radius multiplied by 477, it follows that the sum of the solid angles
of all the distinct cones which can be described with a given point as
vertex, is equal to 477.
484. The solid angles of vertical or opposite cones being equal,
we may infer from what precedes that the sum of the solid angles
of all the complete conical surfaces which can be described with-
out mutual intersection, with a given point as vertex, is equal
to 27r.
485. The solid angle subtended at a point by a superficial area of
any kind, is the solid angle of the cone generated by a straight line
passing through the point, and carried entirely round the boundary
of the area.
486. A very small cone, that is, a cone such that any two posi-
tions of the generating line contain but a very small angle, is said to
be cut at right angles, or orthogonally, by a spherical surface de-
lyo ABSTRACT DYNAMICS.
scribed from its vertex as centre, or by any surface, whether plane or
curved, which touches the spherical surface at the part where the cone
is cut by it.
A very small cone is said to be cut obliquely, when the section is
inclined at any finite angle to an orthogonal section ; and this angle
of inclination is called the obliquity of the section.
The area of an orthogonal section of a very small cone is equal to
the area of an oblique section in the same position, multiplied by the
cosine of the obliquity.
Hence the area of an oblique section of a small cone is equal to
the quotient obtained by dividing the product of the square of its
distance from the vertex, into the solid angle, by the cosine of the
obliquity.
487. Let E denote the area of a very small element of a spherical
surface at the point E (that is to say, an element every part of which
is very near the point E), let w denote the solid angle subtended by E
at any point P, and let PE^ produced if necessary, meet the surface
again in E' \ then a denoting the radius of the spherical surface, we
have
^_2a.i^.PE^
^ " EE' '
For, the obliquity of the element E, considered as a section of the
cone of which P is the vertex and the element
E a section (being the angle between the
given spherical surface and another described
from P as centre, with PE as radius), is
equal to the angle between the radii EP and
EC^ of the two spheres. Hence, by con-
sidering the isosceles triangle ECEf, we find
that the cosine of the obliquity is equal to
hEE' EE'
j-,^ or to ,
EC 2a
and we arrive at the preceding expression for E.
488. The attradioji of a uniform spherical surface on an external
point is the same as if the whole ?nass were collected at the centre^.
Let P be the external point, C the centre of the sphere, and CAP
a straight line cutting the spherical surface in A. Take / in CP,
so that CPf CA, CI may be continual proportionals, and let the
1 This theorem, which is more comprehensive than that of Newton in his first
proposition regarding attraction on an external point (Prop. LXXI.), is fully es-
tablished as a corollary to a subsequent proposition (LXXIII. cor. 2). If we had
considered the proportion of the forces exerted upon two external points at
different distances, instead of, as in the text, investigating the absolute force on
one point, and if besides we had taken together all the pairs of elements which
would constitute two narrow annular portions of the surface, in planes perpen-
dicular to PC, the theorem and its demonstration would have coincided precisely
with Prop. LXXI. of the Pnncipia.
STATICS OF A PARTICLE,— ATTRACTION. 171
whole spherical surface be di-
vided into pairs of opposite ele-
ments with reference to the poijit
L
Let H and H' denote the
magnitudes of a pair of such
elements, situated respectively
at the extremities of a chord
HH' \ and let 00 denote the
magnitude of the solid angle
subtended by either of these elements at the point /.
We have (§ 486),
Zr= 7TT7>5 and II'= — 777777- •
cos CHI cos CHI
Hence, if p denote the density of the surface, the attractions of the
two elements H and H' on F are respectively
CO IH' io IH'
P cos CHI ' FH' ' P cos CHI ' FH" '
Now the two triangles FCH, HCI have a common angle at C, and,
since FC : CH v. CH \ CI^ the sides about this angle are propor-
tional. Hence the triangles are similar; so that the angles CFH
and CHI ^XQ equal, and
IH _CH_ a^
'HF~ CF ~CF'
In the same way it may be proved, by considering the triangles
FCH\ HCI, that the angles CFH and CHIzxq equal, and that
IH _ CH _ _a
HF~ CF ~ CF'
Hence the expressions for the attractions of the elements ^and H
on F become
0) «^ , o) a*
^'^^^CHI' CP' ^ cos CHI' CF' '
which are equal, since the triangle HCH is isosceles; and, for the
same reason, the angles CFH, CFH, which have been proved to
be respectively equal to the angles CHI, CHI, are equal. We
infer that the resultant of the forces due to the two elements is in
the direction FC, and is equal to
a'
20). p. -^-3.
To find the total force on F, we must take the sum of all the
forces along /'C due to the pairs of opposite elements; and, since
the multiplier of w is the same for each pair, we must add all the
values of w, and we therefore obtain (§ 483), for the required re-
sultant,
47rp^^
172 ABSTRACT DYNAMICS.
The numerator of this expression (being the product of the density
into the area of the spherical surface) is equal to the mass of the
entire charge; and therefore the force on P is the same as if the
whole mass were collected at C
Cor. The force on an external point, infinitely near the surface,
is equal to 47rp, and is in the direction of a normal at the point.
The force on an internal point, however near the surface, is, by
a preceding proposition, equal to nothing.
489. Let o- be the area of an infinitely small element of the surface
at any point jP, and at any other point H of
the surface let a small element subtending a
solid angle co, at jP, be taken. The area of
this element will be equal to
\P cos CHP'
and therefore the attraction along HP, which
it exerts on the element <t at P, will be equal
to pi». pa- 0) 2
cos CNP' ^^ cosCBP^"^'
Now the total attraction on the element at T'is in the direction CP;
the component in this direction of the attraction due to the element
H, is
0) . p V ;
and, since all the cones corresponding to the different elements of the
spherical surface lie on the same side of the tangent plane at P, we
deduce, for the resultant attraction on the element o-,
27rp^(r.
From the corollary to the preceding proposition, it follows that this
attraction is half the force which would be exerted on an external
point, possessing the same quantity of matter as the element o-, and
placed infinitely near the surface.
490. In some of the most important elementary problems of the
theory of electricity, spherical surfaces with densities varying inversely
as the cubes of distances from excentric points occur: and it is of
fundamental importance to find the attraction of such a shell on an
internal or external point. This may be done synthetically as follows ;
the investigation being, as we shall see below, virtually the same
as that of § 479, or § 488.
491. Let us first consider the case in which the given point S and
the attracted point P are separated by the spherical surface. The
two figures represent the varieties of this case in which, the point S
being without the sphere, P is within; and, S being within, the
attracted point is external. The same demonstration is applicable
literally with reference to the two figures; but, for avoiding the con-
sideration of negative quantities, some of the expressions may be
conveniently modified to suit the second figure. In such instances
STATICS OF A PARTICLE.— ATTRACTION. 173
the two expressions are given in a double line, the upper being that
which is most convenient for the first figure, and the lower for the
second.
Let the radius of the sphere be denoted by a, and let / be the
distance of S from C, the centre of the sphere (not represented in
the figures).
Join SF and take T in this line (or its continuation) so that
(fig. i) SF.ST=r-a\
(fig. 2) SF.TS=a!'-f\
Through T draw any line cutting the spherical surface at K, K',
Join SK^ SK\ and let the fines so drawn cut the spherical surface
again in E, E'.
Let the whole spherical surface be divided into pairs of opposite
elements with reference to the point T. Let K and K' be a pair of
such elements situated at the extremities of the chord KK\ and
subtending the solid angle m at the point T\ and let elements E and
E' be taken subtending at 6* the same solid angles respectively as the
elements K and K'. By this means we may divide the whole
spherical surface into pairs of conjugate elements, E^ E', since it is
easily seen that when we have taken every pair of elements, K, K\
P
^"
K
the whole surface will have been exhausted, without repetition, by the
deduced elements, E^ E' . Hence the attraction on P will be the
final resultant of the attractions of all the pairs of elements, E^ E'.
Now if p be the surface density at E^ and if i^ denote the attraction
of the element E on F^ we have
E=P^
EF''
According to the given law of density we shall have
X
^~ SE''
where X is a constant. Again, since SEK is equally inclined to the
spherical surface at the two points of intersection, we have
P^SE^ A--— 2ao>.TK\
SK' SX'' KK' '
and hence
X SE' 2ai^.TK^
^ SE''SK'' KK' , 2a TK'
EF* ' KK' ' SE . SK' . EF'
174 ABSTRACT DYNAMICS,
Now, by considering the great circle in which the sphere is cut by a
plane through the line SK, we find that
(fig. i) SK.SE=f'-a\
(fig. 2) KS.SE = a^-f\
and hence SK. SE = ST. ST, firom which we infer that the triangles
XST, TSE are similar; so that TK -. SK v. PE -. ST. Hence
TK^ I
2 CiD2 >
SK\PE^ ST
and the expression for /^becomes
KK''SE.ST'''^'
Modifying this by preceding expressions we have
Similarly, if E' denote the attraction of -£' on T, we have
Now in the triangles which have been shown to be similar, the
angles TKS, ETS are equal; and the same may be proved of the
angles K'ST, TSE'. Hence the two sides SK, SK' of the triangle
KSK' are inclined to the third at the same angles as those between
the line TS and directions TE, TE' of the two forces on the point
T; and the sides SK, SK' are to one another as the forces, E, E',
in the directions TE, TE', It follows, by ' the triangle of forces,'
that the resultant of F and E' is along TS, and that it bears to the
component forces the same ratios as the side KK' of the triangle
bears to the other two sides. Hence the resultant force due to the
two elements E and E' on the point T, is towards 6", and is equal to
KK\f~a^).ST'' ' ' {/'--a') ST' '
The total resultant force will consequently be towards ^S"; and we
find, by summation (§ 466) for its magnitude,
X . 47ra
{r~a')ST''
Hence we infer that the resultant force at any point T, separated
from S by the spherical surface, is the same as if a quantity of
matter equal to '^^^^ were concentrated at the point S,
STATICS OF A PARTICLE.— ATTRACTION. 175
492. To find the attraction when S and F are either both without
or both within the spherical surface.
Take in CS, or in CS produced through S, a point S^, such that
CS. CS^ = a\
Then, by a well-known geometrical theorem, if E be any point on
the spherical surface, we have
SE f
S^E a'
Hence
we have
SE'~/\S,E''
Hence,
if
p being
the electrical density at E,
Xa'
r K
^ S^E' S^E' '
Ai = .
we
have
Hence, by the investigation in the preceding section, the attraction
on F is towards Si , and is the same as if a quantity of matter equal
Xi . A7ra
to T^i 2 were concentrated at that pomt ;
Ii ~ ^
/i being taken to denote CS^. If for /i and X^ we substitute their
values, -p and -^ , we have the modified expression
A -2. . 47ra
for the quantity of matter which we must conceive to be collected
atSy
493. If a spherical surface be electrified in such a way that the
electrical density varies inversely as the cube of the distance from
an internal point S, or from the corresponding external point S-^, it
will attract any external point, as if its whole electricity were con-
176 ABSTRACT DYNAMICS.
centrated at S^ and any internal point, as if a quantity of electricity
greater than its own in the ratio of ^ to /were concentrated at S^
Let the density at E be denoted, as before, by -^^^ . Then, if we
consider two opposite elements at £ and E\ which subtend a solid
angle w at the point S, the areas of these elements being —^ — ^^r, —
and ' ' , the quantity of electricity which they possess will be
££'
\,2a.tti/ I I \ \.2a.(i>
or
2a. W/ I I \
££^\S£'^S£') ^^ S£,S£
Now S£ . SE' is constant (Euc. III. 35) and its value is a^ -/^
Hence, by summation, we find for the total value of electricity on
the spherical surface
Hence, if this be denoted by m^ the expressions in the preceding
paragraphs, for the quantities of electricity which we must suppose to
be concentrated at the point S or *Sj, according as F is without or
within the spherical surface, become respectively
nif and ^ m.
494. The direct analytical solution of such problems consists in
the expression, by § 408, of the three components of the whole at-
traction as the sums of its separate parts due to the several particles
of the attracting body; the transformation, by the usual methods, of
these sums into definite integrals; and the evaluation of the latter.
This is, in general, inferior in elegance and simplicity to the less
direct mode of solution depending upon the determination of the
potential energy of the attracted particle with reference to the forces
exerted upon it by the attracting body, a method which we shall
presently develop with peculiar care, as it is of incalculable value in
the theories of Electricity and Magnetism as well as in that of
Gravitation. But before we proceed to it, we give some instances of
the direct method.
{a) A useful case is that of the attraction of a circular plate of
uniform surface density on a point in a line through its centre, and
perpendicular to its plane.
All parallel slices, of equal thickness, of any cone attract equally
(both in magnitude and direction) a particle at the vertex.
For the proposition is true of a cone of infinitely small angle, the
masses of the slices being evidently as the squares of their distances
from the vertex. If / be the thickness, p the volume density, and w
the angle, the attraction is w/p.
All slices of a cone of infinitely small angle, if of equal thickness
STATICS OF A PARTICLE.— ATTRACTION. 177
and equally inclined to the axis of the cone, exert equal forces on a
particle at the vertex. For the area of any inclined section, whatever
be its orientation, is greater than that of the corresponding transverse
section in the ratio of unity to the cosine of the angle of inclination.
Hence if a plane touch a sphere at a
point B, and if the plane and sphere have
equal surface density at corresponding
points P and / in a line drawn through
A^ the point diametrically opposite to By
corresponding elements at P and / exert
equal attraction on a particle at A.
Thus the attraction on A, of any part
of the plane, is the same as that of the
corresponding part of the sphere, cut out
by a cone of infinitely small angle whose vertex is A.
Hence if we resolve along the line AB the attraction of pq on A,
the component is equal to the attraction along Ap of the -transverse
section pr, i.e. /qw, where co is the angle subtended at A by the
element pq, and p the surface density.
Thus any portion whatever of the sphere attracts A along AB
with a force proportional to its spherical opening as seen from ^;
and the same is, by what was proved above, true of a flat plate.
Hence as a disc of radius a subtends at a point distant h from it,
in the direction of the axis of the disc, a solid angle
\ J/i' + aV'
the attraction of such a disc is
''"'{'- JW^)'
which for an infinite disc becomes, whatever the distance h^
27: p.
From the preceding formula many useful results may easily be
deduced : thus,
{b) A uniform cylinder of length /, and diameter a, attracts a point
in its axis at a distance x from the nearest end with a force
27rp {/- J{x + lY + a' + J^^T?].
When the cylinder is of infinite length (in one direction) the at-
traction is therefore
27rp [Jx^ +a^ -x);
and, when the attracted particle is in contact with the centre of the
end of the infinite cylinder, this is
2-7? pa.
(c) A right cone, of semivertical angle a, and length /, attracts a
T. 12
178 ABSTRA CT D YNAMICS.
particle at its vertex. Here we have at once for the attraction, the
expression
2irpl{l -COS a),
which is simply proportional to the length of the axis.
It is of course easy, when required, to find the necessarily less
simple expression for the attraction on any point of the axis.
(d) For magnetic and electro-magnetic applications a very useful
case is that of two equal uniform discs, each perpendicular to the line
joining their centres, on any point in that line — their masses (§ 478)
being of opposite sign — that is, one repelling and the other attracting.
Let a be the radius, p the mass of a superficial unit, of either, c
their distance, x the distance of the attracted point from the nearest
disc, The whole force is evidently
{X + ^ X ^
J{x + cy + a'~ Jl^^T^j'
In the particular case when c is diminished without limit, this
becomes
27rp{:
(^ + «^)l
495. Let P and P' be two points infinitely near one another on
two sides of a surface over which matter is distributed ; and let p be
the density of this distribution on the surface in the neighbourhood
of these points. Then whatever be the resultant attraction, i?, at /*,
due to all the attracting matter, whether lodging on this surface, or
elsewhere, the resultant force, R\ on P' is the resultant of a force
equal and parallel to R, and a force equal to 4Trp, in the direction
from P' perpendicularly towards the surface. For, suppose PP' to
be perpendicular to the surface, which will not limit the generality of
the proposition, and consider a circular disc, of the surface, having
its centre in Pp\ and radius infinitely small in comparison with the
radii of curvature of the surface but infinitely great in comparison
with PjP\ This disc will [§ 494] attract -P and P with forces,
each equal to 27rp and opposite to one another in the line FF'.
Whence the proposition. It is one of much importance in the theory
of electricity.
496. It may be shown that at the southern base of a hemispherical
hill of radius a and density p, the true latitude (as measured by the
aid of the plumb-line, or by reflection of starlight in a trough of
mercury) is diminished by the attraction of the mountain by the
angle
G-ipa^
where G is the attraction of the earth, estimated in the same units.
STATICS OF A PARTICLE,— ATTRACTION. 179
Hence, if R be the radius and o- the mean density of the earth, the
angle is
^Trpa ^ pa . ,
^.<rli-%pa '°'i^ approximately.
Hence the latitudes of stations at the base of the hill, north and
south of it, diifer by ^ ^2 + — h instead of by ^, as they would
do if the hill were removed.
In the same way the latitude of a place at the southern edge of a
hemispherical caz'ify is increased on account of the cavity by ^ ^-=
where p is the density of the superficial strata.
497. As a curious additional example of the class of questions
we have just considered, a deep crevasse, extending east and west,
increases the latitude of places at its southern edge by (approx-
imately) the angle f ^ where 0 is the density of the crust of the
CTjCV
earth, and a is the width of the crevasse. Thus the north edge of
the crevasse will have a lower latitude than the south edge if f - > i,
which might be the case, as there are rocks of density f x 5*5 or
3 '67 times that of water. At a considerable depth in the crevasse,
this change of latitudes is nearly doubled^ and then the southern side
has the greater latitude if the density of the crust be not less than
1*83 times that of water.
498. It is interesting, and will be useful later, to consider as a
particular case, the attraction of a sphere whose mass is composed of
concentric layers, each of uniform density. Let <r be, as above, the
mean density of the whole globe, and t the density of the upper crust.
The attraction at a depth ^, small compared with the radius, is
^i:<T^{R-h)^G,
where otj denotes the mean density of the nucleus remaining when a
shell of thickness h is removed from the sphere. Also, evidently,
^i?<T^ {R - Hf + 47rT (i? - h^h = Itto-J?',
or G,(R-/iy + 4'^T{R-/iy/i=:GR%
whence G^ = G(i + -^j-47rTk
The attraction is therefore unaltered at a depth /i if
_=*,r(T=27rT, I.e. T = |cr.
499. Some other simple cases may be added here, as their results
will be of use to us subsequently.
12^2
i8o
ABSTRACT DYNAMICS.
(a) The attraction of a circular arc, AB, of uniform density, on a
particle at the centre, C, of the circle, lies
evidently in the line CD bisecting the arc.
Also the resolved part parallel to CD of
the attraction of an element at F is
mass of element at /* „^ ^
^7^2 cos . z BCD.
Now suppose the density of the chord AB
to be the same as that of the arc. Then
for (mass of element at F x cos l BCD)
we may put (mass of projection of element
on AB at 0; since, if BT be the tan-
gent at B, I BTQ = L BCD,
Hence attraction along CD =
sum of projected elements
CD^
pAB
CD"'
if p be the density of the given arc,
_ 2psin L ACD
CD
It is therefore the same as the attraction of a mass equal to the
chord, with the arc's density, concentrated at the point D.
(p) Again, a limited straight line of uniform density attracts any
external point in the
\C same direction and with
the same force as the
corresponding arc of a
circle of the same den-
sity, which has the point
for centre, and touches
the straight line.
For if CpB be drawn
cutting the circle in p
and the line in B; ele-
CB
ment at p : element Sit B :: Cp : CB ^^; that is, as Cp' : CB\
Hence the attractions of these elements on C are equal and in the
same line. Thus the arc ab attracts C as the line AB does ; and, by
the last proposition, the attraction oi AB bisects the angle -^C5, and
is equal to
-^sin^z^C^.
STATICS OF A PARTICLE.— ATTRACTION. i8i
{c) This may be
put into other use-
ful forms — thus, let
CKF bisect the an-
gle ACB, and let
Aa^ Bb, EF^ be
drawn perpendicular
to CF from the
ends and middle
point of AB, We
have
. „_^ KB . -__- AB CD
sm L KCB^-^sm l CKD^-j^-^^-^,
Hence the attraction, which is along CK, is
2pAB pAB
CF..
(I).
(AC+ CB) CK 2 (^C+ CB) (AC-^r CB'-AB')
For evidently,
bK : Ka :: BK : KA :: BC : CA :: bC : (G?,
i.e., ^/^ is divided, externally in C, and internally in X, in the same
ratio. Hence, by geometry,
XC. CF=aC. Cb = i{A C+ CB' - AB'},
which gives the transformation in (i).
(d) CF is obviously the tangent at C to a hyperbola, passing
through that point, and having A and B as foci. Hence, if in
any plane through AB any hyperbola be described, with foci A
and Bj it will be a line of force as regards the attraction of the
line AB ; that is, as will be more fully explained later, a curve which
at every point indicates the direction of attraction.
(e) Similarly, if a prolate spheroid be described with foci A and B,
and passing through C, Ci^will evidently be the normal at C; thus
the force on a particle at C will be perpendicular to the spheroid ;
and the particle would evidently rest in equilibrium on the surface,
even if it were smooth. This is an instance of (what we shall pre-
sently develop at some length) a surface of equilibrium, a level
surface, or an equipotential surface.
(/) We may further prove, by a simple application of the
preceding theorem, that the lines of force due to the attraction
of two infinitely long rods in the line AB produced, one of which
is attractive and the other repulsive, are the series of ellipses
described from the extremities, A and B, as foci, while the surfaces
of equilibrium are generated by the revolution of the confocal
hyperbolas.
i83 , ABSTRACT DYNAMICS.
500. As of immense importance, in the theory not only of gra-
vitation but of electricity, of magnetism, of fluid motion, of the
conduction of heat, etc., we give here an investigation of the most
important properties of the Potential.
501. This function was introduced for gravitation by Laplace,
but the name was first given to it by Green, who may almost be
said to have created the theory, as we now have it. Green's work
was neglected till 1846, and before that time most of its important
theorems had been re-discovered by Gauss, Chasles, Sturm, and
Thomson.
In § 245, the potential energy of a conservative system in any con-
figuration was defined. When the forces concerned are forces acting,
either really or apparently, at a distance, as attraction of gravitation,
or attractions or repulsions of electric or magnetic origin, it is in
general most convenient to. choose, for the zero configuration, infinite
distance between the bodies concerned. We have thus the following
definition : —
502. The mutual potential energy of two bodies in any relative
position is the amount of work obtainable from their mutual repulsion,
by allowing them to separate to an infinite distance asunder. When
the bodies attract mutually, as for instance when no other force than
gravitation is operative, their mutual potential energy, according to
the convention for zero now adopted, is negative, or their exhaustion
of potential energy is positive.
503. The Potential at any point, due to any attracting or repelling
body, or distribution of matter, is the mutual potential energy between
it and a unit of matter placed at that point. But in the case of
gravitation, to avoid defining the potential as a negative quantity,
it is convenient to change the sign. Thus the gravitation potential,
at any point, due to any mass, is the quantity of work required to
remove a unit of matter from that point to an infinite distance.
504. ^ Hence, if V be the potential at any point P, and V^ that at
a proximate point (2, it evidently follows from the above definition
that V- Fj is the work required to remove an independent unit of
matter from P\.q Q-, and it is useful to note that this is altogether
independent of the form of the path chosen between these two points,
as it gives us a preliminary idea of the power we acquire by the
introduction of this mode of representation.
Suppose Q to be so near to P that the attractive forces exerted on
unit of matter at these points, and therefore at any point in the line
jP^, may be assumed to be equal and parallel. Then if F represent
the resolved part of this force along PQ, F. PQ is the work required
to transfer unit of matter from P to Q. Hence
V-V^=F.PQ,
""—PQ^
STATICS OF A PARTICLE.— ATTRACTION. 183
that is, the attraction on unit of matter at P in any direction PQ^
is the rate at which the potential at P increases per unit of length
oiPQ.
505. A surface, at every point of which the potential has the same
value, and therefore called an Equipotential Surface^ is such that the
attraction is everywhere in the direction of its normal. For in no
direction along the surface does the potential change in value, and
therefore there is no force in any such direction. Hence if the
attracted particle be placed on such a surface (supposed smooth and
rigid), it will rest in any position, and the surface is therefore some-
times called a Surface of Eqicilibrium. We shall see later, that the
force on a particle of a liquid at the free surface is always in the
direction of the normal, hence the term Level Surface, which is often
used for the other terms above.
506. If a series of equipotential surfaces be constructed for values
of the potential increasing by equal small amounts, it is evident from
§ 504 that the attraction at any point is inversely proportional to
the normal distance between two successive surfaces close to that
point; since the numerator of the expression for F is, in this case,
constant.
507. A line drawn from any origin, so that at every point of its
length its tangent is the direction of the attraction at that point, is
called a Line of Force; and it obviously cuts at right angles every
equipotential surface which it meets.
These three last sections are true whatever be the law of attraction ;
in the next we are restricted to the law of the inverse square of the
distance.
508. If, through every point of the boundary of an infinitely
small portion of an equipotential surface, the corresponding lines of
force be drawn, we shall evidently have a tubular surface of infinitely
small section. The resultant force, being at every point tangential
to the direction of the tube, is inversely as its normal transverse
section.
This is an immediate consequence of a most important theorem,
which will be proved later. The surface integral of the attraction
exerted by any distribution of matter i?t the direction of the normal at
every poi7it of any closed suiface is ^irM; where M is the amount of
matter withiii the surface., while the attraction is considered positive or
negative according as it is inibards or outwards at any poi?tt of the
surface.
For in the present case the force perpendicular to the tubular
part of the surface vanishes, and we need consider the ends only.
VVhen none of the attracting mass is within the portion of the tube
considered, we have at once
F^uT - Fvs' = o,
i^ being the force at any point of the section whose area is zsr.
This is equivalent to the celebrated equation of Laplace.
i84 ABSTRACT DYNAMICS.
When the attracting body is symmetrical about a point, the lines
of force are obviously straight lines drawn from this point. Hence
the tube is in this case a cone, and, by § 486, ra- is proportional to
the square of the distance from the vertex. Hence F is inversely
as the square of the distance for points external to the attracting
mass.
When the mass is symmetrically disposed about an axis in in-
finitely long cylindrical shells, the lines of force are evidently perpen-
dicular to the axis. Hence the tube becomes a wedge, whose section
is proportional to the distance from the axis, and the attraction is
therefore inversely as the distance from the axis.
When the mass is arranged in infinite parallel planes, each of
uniform density, the lines of force are obviously perpendicular to
these planes; the tube becomes a cylinder; and, since its section is
constant, the force is the same at all distances.
If an infinitely small length / of the portion of the tube considered
pass through matter of density p, and if w be the area of the section
of the tube in this part, we. have
F'us — F'w = 47r/o)p,
This is equivalent to Poisson's extension of Laplace's equation.
509. In estimating work done against a force which varies in-
versely as the square of the distance from a fixed point, the mean
force is to be reckoned as the geometrical mean between the forces
at the beginning and end of the path: and, whatever may be the
path followed, the effective space is to be reckoned as the difference
of distances from the attracting point. Thus the work done in any
course is equal to the product of the difference of distances of the
extremities from the attracting point, into the geometrical mean of
the forces at these distances; or, if O be the attracting point, and 7/1
its force on a unit mass at unit distance, the work done in moving a
particle, of unit mass, from any position F to any other position
F\ is
n' mm
(,OP-OP) J
or
Qp2 Qp,2^ -- Qp Qp,'
To prove this it is only necessary to remark, that for any infinitely
small step of the motion, the effective space is clearly the difference
of distances from the centre, and the working force may be taken as
the force at either end, or of any intermediate value, the geometrical
mean for instance: and the preceding expression applied to each
infinitely small step shows that the same rule holds for the sum
making up the whole work done through any finite range, and by
any path.
Hence, by § 503, it is obvious that the potential at F, of a mass m
m
situated at (9, is -7^] and thus that the potential of any mass at a
point F is to be found by adding the quotients of every portion of
the mass, each divided by its distance from /*.
STATICS OF A PARTICLE.— ATTRACTION. 185
510. Let ^ be any dosed surface, and let 6^ be a point, either
external or internal, where a mass, m, of matter is collected. Let N
be the component of the attraction of m in the direction of the
normal drawn inwards from any point P, of S. Then, if da denotes
an element of 6", and // integration over the whole of it,
// N d<T = 47rw, or = o,
according as O is internal or external.
Case /, O internal. Let OP^PJP^... be a straight line drawn in
any direction from O, cutting 6* in P^^ P^, P^, etc., and therefore
passing out at P^, in at P^^ out again at P^, in again at P^, and so
on. Let a conical surface be described by lines through O, all in-
finitely near OP^P„..., and let w be its solid angle (§ 482). The
portions of jjNda- corresponding to the elements cut from S by
this cone will be clearly each equal in absolute magnitude to lonij
but will be alternately positive and negative. Hence as there is an
odd number of them, their sum is + irnn. And the sum of these, for
all solid angles round O, is (§ 483) equal to ^irm; that is to say,
jjNda-^ 4irm.
Case II, O external. Let OP^P^P^... be aline drawn from O pass-
ing across S, inwards at P^, outwards at P„, and so on. Drawing,
as before, a conical surface of infinitely small solid angle, w, we have
still uitn for the absolute value of each of the portions of jjNdcr
corresponding to the elements which it cuts from S', but their signs
are alternately negative and positive ; and therefore as their number
is even, their sum is zero. Hence
jjNdiT = 0.
From these results it follows immediately that if there be any con-
tinuous distribution of matter, partly within and partly without a
closed surface S, and N and da- be still used with the same signifi-
cation, we have
J5Nd(T = 4TrM
if M denote the whole amount of matter within S,
511. From this it follows that the potential cannot have a maxi-
mum or minimum value at a point in free space. For if it were so,
a closed surface could be described about the point, and indefinitely
near it, so that at every point of it the value of the potential would be
less than, or greater than, that at the point ; so that N would be
negative or positive all over the surface, and therefore jjNda- would
be finite, which is impossible, as the surface contains none of the
attracting mass.
512. It is also evident that iVmust have positive values at some
parts of this surface, and negative values at others, unless it is zero
all over it. Hence in free space the potential, if not constant round
any point, increases in some directions from it, and diminishes in
i86 ABSTRACT DYNAMICS.
others; and therefore a material particle placed at a point of
zero force under the action of any attracting bodies, and free
from all constraint, is in unstable equilibrium, a result due to
Earnshaw\
513. If the potential be constant over a closed surface which
contains none of the attracting mass, it has the same constant value
throughout the interior. For if not, it must have a maximum or
minimum value somewhere within, which is impossible.
514. The mean potential over any spherical surface, due to matter
entirely without it, is equal to the potential at its centre ; a theorem
apparently first given by Gauss. See also Cambridge Mathematical
Journal^ Feb. 1845 (vol. iv. p. 225). This proposition is merely an
extension, to any masses, of the converse of the following statement,
which is easily seen to follow from the results of §§ 479, 488 expressed
in potentials instead of forces. The potential of an uniform spherical
shell at an external point is the same as if its mass were condensed
at the centre. At all internal points it has the same value as at the
surface.
515. If the potential of any masses has a constant value, F,
through any finite portion, K, of space, unoccupied by matter, it is
equal to F through every part of space which can be reached in any
way without passing through any of those masses : a very remarkable
proposition, due to Gauss. For, if the potential difier from V in
space contiguous to K, it must (§513) be greater in some parts and
less in others.
From any point C within K, as centre, in the neighbourhood of a
place where the potential is greater than V, describe a spherical
surface not large enough to contain any part of any of the attracting
masses, nor to include any of the space external to K except such
as has potential greater than V. But this is impossible, since we
have just seen (§ 514) that the mean potential over the spherical
surface must be V. Hence the supposition that the potential is
greater than V in some places and less in others, contiguous to K
and not including masses, is false.
516. Similarly we see that in any case of symmetry round an
axis, if the potential is constant through a certain finite distance,
however short, along the axis, it is constant throughout the whole
space that can be reached from this portion of the axis, without
crossing any of the masses.
517. Let 5 be any finite portion of a surface, or complete closed
surface, or infinite surface, and let E be any point on S. (a) It is
possible to distribute matter over S so as to produce potential equal
to T {£), any arbitrary function of the position of £, over the whole
of S. {b) There is only one whole quantity of matter, and one
distribution of it, which can satisfy this condition. For the proof of
^ Cambridge Phil. Trans., March, 1839.
STATICS OF A PARTICLE.—ATTRACTION. 187
this and of several succeeding theorems, we refer the reader to our
larger work.
518. It is important to remark that, if S consist, in part, of a
closed surface, <2) the determination of U^ the potential at any point,
within it will be independent of those portions of 6", if any, which
lie without it; and, vice versa, the determination of C/ through external
space will be independent of those portions of S, if any, which lie
within the part Q. Or if S consist, in part, of a surface Q, extend-
ing infinitely in all directions, the determination of C/ through all
space on either side of Q, is independent of those portions of S, if
any, which lie on the other side.
519. Another remark of extreme importance is this: — If F(E)
be the potential at E of any distribution, M, of matter, and if S be
such as to separate perfectly any portion or portions of space, ZT,
from all of this matter; that is to say, such that it is impossible to
pass into ZTfrom any part of J/ without crossing -5"; then, throughout
JI, the value of U will be the potential of M.
520. Thus, for instance, if .S consist of three detached surfaces,
5j, S^, 6*3, as in the diagram, of which S^, S^ are closed, and S^ is
an open shell, and if F {E) be
the potential due to M, at any
point, E, of any of these portions
of S', then throughout H^ and
Z^, the spaces within S^ and witli- \^ f ^^ ^^^^( //< W //,
out 6*2, the value of U is simply
the potential of M. The value of
U through K, the remainder of
space, depends, of course, on the
character of the composite sur-
face S.
521. From § 518 follows the grand proposition : — // is possible to
fitid one, but 710 other thati one, distribution of matter over a surface S
which shall produce over S, and throughout all space H separated by S
from every part of M, the same potential as any given mass M.
Thus, in the preceding diagram, it is possible to find one, and but
one, distribution of matter over S^, S^, S.^ which shall produce over
6*3 and through H^ and H^ the same potential as M.
The statement of this proposition most commonly made is : // is
possible to distribute matter over any surface, S, completely enclosing a
mass M, so as to produce the same potential as M through all space
outside M ; which, though seemingly more limited, is, when inter-
preted with proper mathematical comprehensiveness, equivalent to the
foregoing.
522. If S consist of several closed or infinite surfaces, S^, S^, S^,
respectively separating certain isolated spaces H^, H^, H^, from H,
the remainder of all space, and if F {E) be the potential of masses
m^, m^, m^, lying in the spaces H^, H^, H^\ the portions of C/"due to
i88
ABSTRACT DYNAMICS,
Sxt -5*2, ^g, respectively will throughout H be equal respectively to
the potentials oi m^^ m^, m^, separately.
For, as we have just seen, it is possible to find one, but only one,
distribution of matter over S, which shall produce the potential oi m^^
jj throughout all the space H, H^,
j^g, etc., and one, but only one,
distribution over S^ which shall
produce the potential of m^
throughout ZT, H^, H^, etc. ; and
so on. But these distributions
on ^Sj, vSg, etc., jointly constitute
a distribution producing the po-
tential F {E) over every part of
S^ and therefore the sum of the
potentials due to them all, at any point, fulfils the conditions pre-
sented for U. This is therefore (§ 5 18) the solution of the problem.
523. Considering still the case in which F {E) is prescribed to be
the potential of a given mass, M : let S be an equipotential surface
enclosing M, or a group of isolated surfaces enclosing all the parts of
M^ and each equipotential for the whole of M. The potential due to
the supposed distribution over S will be the same as that of J/",
through all external space, and will be constant (§ 514) through each
enclosed portion of space. Its resultant attraction will therefore be
the same as that of M on all external points, and zero on all internal
points. Hence we see at once that the density of the matter dis-
tributed over it, to produce F (F), is equal to — where F denotes
4Tr
the resultant force of M, at the point F.
524. When M consists of two portions m^ and pi separated by an
equipotential S^, and 6* consists of two portions, S^ and S\ of which
the latter separates the former perfectly from m' ; we see, by § 522,
that the distribution over S^ produces through all space on the side
of it on which S' lies, the same potential, P\, as m^, and the dis-
tribution on S' produces through space on the side of it on which S^
lies, the same potential, F', as ?//. But the supposed distribution on
the whole of 6" is such as to produce a -constant potential, Ci, over S^,
STATICS OF A PARTICLE,— ATTRACTION. 189
and consequently the same at every point within S^. Hence the in-
ternal potential, due to S^ alone, is C^ - V.
Thus, passing from potentials to attractions, we see that the re-
sultant attraction of 6*1 alone, on all points on one side of it, is the
same as that of m^ ; and on the other side is equal and opposite to
that of the remainder m' of the whole mass. The most direct and
simple complete statement of this result is as follows : —
If masses m, 7n\ in portions of space, H^ II\ completely separated
from one another by one continuous surface S, whether closed or
infinite, are known to produce tangential forces equal and in the
same direction at each point of S, one and the same distribution
of matter over S will produce the force of m throughout II\ and
that of m' throughout II. The density of this distribution is equal to
— , if i? denote the resultant force due to one of the masses, and
the other with its sign changed. And it is to be remarked that the
direction of this resultant force is, at every point, E, of S, perpen-
dicular to S, since the potential due to one mass, and the other with
its sign changed, is constant over the whole of S.
525. Green, in first publishing his discovery of the result stated
in § 523, remarked that it shows a way to find an infinite variety of
closed surfaces for any one of which we can solve the problem of
determining the distribution of matter over it which shall produce
a given uniform potential at each point of its surface, and con-
sequently the same also throughout its interior. Thus, an example
which Green himself gives, let J/ be a uniform bar of matter, AA\
The equipotential surfaces round it are, as we have seen above
(§ 499 W)j prolate ellipsoids of revolution, each having A and A' for
its foci; and the resultant force at Cwas found to be
^ CR
the whole mass of the bar being denoted by m^ its length by 2dJ, and
^'C+^Cby 2/. We conclude that a distribution of matter over
the surface of the ellipsoid, having
I m.CF
for density at C, produces on all external space the same resultant
force as the bar, and zero force or a constant potential through the
internal space. This is a particular case of the general result re-
garding ellipsoidal shells, proved below, in §§ 536, 537.
526. As a second example, let J/" consist of two equal particles,
at points /, /'. If we take the mass of each as unity, the potential at
Fisj-p + -jTp'} and therefore
IP^ FP ^
190
ABSTRACT DYNAMICS.
is the equation of an equipotential surface ; it being understood that
negative values of IP and I'P are inadmissible, and that any con-
stant value, from 00 to o, may be given to C. The curves in the
annexed diagram have t)een drawn, from this equation, for the cases
of C equal respectively to 10, 9, 8, 7, 6, 5, 4-5, 4-3, 4-2, 4*1, 4, 3-9,
3*8, 37, 3*5, 3, 2-5, 2; the value of //'being unity.
The corresponding equipotential surfaces are the surfaces traced
by these curves, if the whole diagram is made to rotate round II' as
axis. Thus we see that for any values of C less than 4 the equi-
potential surface is one closed surface. Choosing any one of these
surfaces, let R denote the resultant of forces equal to ^^ and -frSi
in the lines PI and PI'. Then if matter be distributed over this
surface, with density at P equal to — , its attraction on any internal
4t
point will be zero; and on any external point, will be the same as
that of /and/'.
527. For each value of C greater than 4, the equipotential surface
consists of two detached ovals approximating (the last three or four
in the diagram, very closely) to spherical surfaces, with centres lying
between the points / and /', but approximating more and more
closely to these points, for larger and larger values of C.
Considering one of these ovals alone, one of the series enclosing
/', for instance, and distributing matter over it according to the same
r>
law of density, — , we have a shell of matter which exerts (§ 525)
47r
E
STATICS OF A PARTICLE.- ATTRACTION. 191
on external points the same force as /'; and on internal points a
force equal and opposite to that of /.
528. As an example of exceedingly great importance in the theory
of electricity, let M consist of a positive mass, m, concentrated
at a point /, and a negative
mass, — ;//, at /' j and let ^S"
be a spherical surface cutting
//' and //' produced in points
A, A^, such that
lA :AI'::IA, iTA/.-.m : m'.
Then, by a well-known geo-
metrical proposition, we shall
have m : T£ v. m -. m' \ and
therefore
m _ in
lE~ TE'
Hence, by what we have just seen, one and the same distribution of
matter over »S will produce the same force as m! through all external
space, and the same as m through all the space within S. And,
ffi . in'
finding the resultant of the forces y^ in EI^ and -77^^ in I'E^ pro-
duced, which, as these forces are inversely as IE to I'E^ is (§222)
equal to
m j.^, m^ir I
IE\I'E ' °^ 1^ IE'' '
we conclude that the density in the shell at E is
m^ir I
47rw' ' lE^ '
That the shell thus constituted does attract external points as if its
mass were collected at /', and internal points as a certain mass col-
lected at /, was proved geometrically in § 491 above.
529. If the spherical surface is given, and one of the points, /, /',
CA^
for instance /, the other is found by taking CI' = -?^ ; and for the
mass to be placed at it we have
TA CA cr
'^='^AI = '^-Cl = ''' CA-
Hence, if we have any number of particles m^^ m^^ etc., at points
/j, /g, etc., situated without S, we may find in the same way cor-
responding internal points /'„ I\, etc., and masses m\^ m\, etc. ;
and, by adding the expressions for the density at E given for each
pair by the preceding formula, we get a spherical shell of matter
which has the property of acting on all external space with the same
force as —in\, -^'25 ^tc, and on all internal points with a force
equal and opposite to that of Wj, m^, etc.
192 ABSTRACT DYNAMICS.
530. An infinite number of such particles may be given, con-
stituting a continuous mass M\ when of course the corresponding
internal particles will constitute a continuous mass, —M'^ of the
opposite kind of matter; and the same conclusion will hold. If S is
the surface of a solid or hollow metal ball connected with the earth
by a fine wire, and M an external influencing body, the shell of matter
we have determined is precisely the distribution of electricity on S
called out by the influence of M: and the mass - M\ determined as
above, is called the Electric Image of M in the ball, since the electric
action through the whole space external to the ball would be
unchanged if the ball were removed and — M' properly placed in
the space left vacant. We intend to return to this subject under
Electricity.
531. Irrespectively of the special electric application, this method
of images gives a remarkable kind of transformation which is often
useful. It suggests for mere geometry what has been called the
transformation by reciprocal radius-vectors ; that is to say, the sub-
stitution for any set of points, or for any diagram of lines or surfaces,
another obtained by drawing radii to them from a certain fixed point
or origin, and measuring off lengths inversely proportional to these
radii along their directions. We see in a moment by elementary
geometry that any line thus obtained cuts the radius-vector through
any point of it at the same angle and in the same plane as the line
from which it is derived. Hence any two lines or surfaces that cut
one another give two transformed lines or surfaces cutting at the
same angle : and infinitely small lengths, areas, and volumes trans-
form into others whose magnitudes are altered respectively in the
ratios of the first, second, and third powers of the distances of the
latter from the origin, to the same powers of the distances of the
former from the same. Hence the lengths, areas, and volumes in
the transformed diagram, corresponding to a set of given equal
infinitely small lengths, areas, and volumes, however situated, at
different distances from the origin, are inversely as the squares, the
fourth powers and the sixth powers of these distances. Further, it is
easily proved that a straight line and a plane transform into a circle
and a spherical surface, each passing through the origin ; and that,
generally, circles and spheres transform into circles and spheres.
532. In the theory of attraction, the transformation of masses,
densities, and potentials has also to be considered. Thus, according
to the foundation of the method (§ 530), equal masses, of infinitely
small dimensions at different distances from the origin, transform into
masses inversely as these distances, or directly as the transformed
distances : and, therefore, equal densities of lines, of surfaces, and of
solids, given at any stated distances from the origin, transform into
densities directly as the first, the third, and the fifth powers of those
distances ; or inversely as the same powers of the distances, from
the origin, of the corresponding points in the transformed system.
The usefulness of this transformation in the theory of electricity,
STATICS OF A PARTICLE.— ATTRACTION. 193
and of attractian in general, depends entirely on the following
theorem : —
Let <f> denote the potential at F due to the given distribution, and <^'
the potential at F' due to the transformed distribution : then shall
<i> = - <^ = - ^.
a r
Let a mass m collected at / be any part of the given distri-
bution, and let m at /' be
the corresponding part in y^
the transformed distribution, -^
We have / \
a'=Or.OI^OF\OF, /
and therefore /
01: OFv. OF' : OF; ^
which shows that the triangles //^(9, /*'/'(? are similar, so that
IF iFF:: J 01 OF : JOF\Or - OIOF : a\
We have besides
m : m \ : 01 : a,
and therefore
mm
Jp--rF.::a:OP.
Hence each term of <^ bears to the corresponding term of <^'
the same ratio; and therefore the sum, </>, must be to the sum,
<^', in that ratio, as was to be proved.
533. As an example, let the given distribution be confined to a
spherical surface, and let O be its centre and a its own radius. The
transformed distribution is the same. But the space within it becomes
transformed into the space, without it. Hence if ^ be the potential
due to any spherical shell at a point P, within it, the potential due
8
to the same shell at the point F in OF produced till OF' = -y^ , is
equal to yrpt ^ (which is an elementary proposition in the spherical
harmonic treatment of potentials, as we shall see presently). Thus,
for instance, let the distribution be uniform. Then, as we know
there is no force on an interior point, <^ must be constant; and
therefore the potential at F\ any external point, is inversely propor-
tional to its distance from the centre.
Or let the given distribution be a uniform shell, 5, and let O be
any eccentric or any external point. The transformed distribution
becomes (§§ 531, 532) a spherical shell, 6", with density varying
inversely as the cube of the distance from O. If O is within 6", it is
also enclosed by .S", and the whole space within S transforms into
T. 13
194 ABSTRACT DYNAMICS.
the whole space without S'. Hence (§ 532) the potential of S' at
any point without it js inversely as the distance from 0^ and is there-
fore that of a certain quantity of matter collected at O. Or if O is
external to »S, and consequently also external to S\ the space within
S transforms into the space within S' . Hence the potential of S'
at any point within it is the same as that of a certain quantity of
matter collected at (9, which is now a point external to it. Thus,
without taking advantage of the general theorems (§§ 517, 524), we
fall back on the same results as we inferred from them in § 528, and
as we proved synthetically earlier (§§ 488, 491, 492). It may be
remarked that those synthetical demonstrations consist merely of
transformations of Newton's demonstration, that attractions balance
on a point within a uniform shell. Thus the first of them (§ 488)
is the image of Newton's in a concentric spherical surface ; and the
second is its image in a spherical surface having its centre external
to the shell, or internal but eccentric, according as the first or the
second diagram is used.
534. We shall give just one other application of the theorem
of § 532 at present, but much use of it will be made later in the
theory of Electricity.
Let the given distribution of matter be a uniform solid sphere, j5,
and let O be external to it. The transformed system will be a solid
sphere, B\ with density varying inversely as the fifth power of the
distance from (9, a point external to it. The potential of B is the
same throughout external space as that due to its mass, w, collected
at its centre, C. Hence the potential of B' through space external
to it is the same as that of the corresponding quantity of matter
collected at C\ the transformed position of C. This quantity is of
course equal to the mass of B'. And it is easily proved that C is
the position of the image of O in the spherical surface of B\ We
conclude that a solid sphere with density varying inversely as the
fifth power of the distance from an external point, O, attracts any
external point as if its mass were condensed at the image of O in its
external surface. It is easy to verify this for points of the axis by
direct integration, and thence the general conclusion follows ac-
cording to § 508.
535. The determination of the attraction of an ellipsoid, or of an
eUipsoidal shell, is a problem of great interest, and its results will be
of great use to us afterwards, especially in Magnetism. We have
left it till now, in order that we may be prepared to apply the pro-
perties of the potential, as they afford an extremely elegant method
of treatment. A few definitions and lemmas are necessary.
Corresponding points on two confocal ellipsoids are such as coincide
when either ellipsoid by a pure strain is deformed so as to coincide
with the other.
And it is easily shown, that if any two points, /*, Q, be assumed on
one shell, and their corresponding points, /, ^, on the other, we have
Pq-Qp.
STATICS OF A PARTICLE.—ATTRACTION. 195
The species of shell which it is most convenient to employ in the
subdivision of a homogeneous ellipsoid is bounded by similar, simi-
larly situated, and concentric ellipsoidal surfaces; and it is evident
from the properties of pure strain (§ 141) that such a shell may be
produced from a spherical shell of uniform thickness by unijform
extensions and compressions in three rectangular directions. Unless
the contrary be specified, the word 'shell' in connexion with this
subject will always signify an infinitely thin shell of the kind now
described.
536. Since, by § 479, a homogeneous spherical shell exerts no
attraction on an internal point, a homogeneous shell (which need
not be infinitely thin) bounded by similar, and similarly situated, and
concentric ellipsoids, exerts no attraction on an internal point.
For suppose the spherical shell of § 479, by simple extensions and
compressions in three rectangular directions, to be transformed into
an ellipsoidal shell. In this distorted form the masses of all parts
are reduced or increased in the proportion of the mass of the eUipsoid
to that of the sphere. Also the ratio of the lines HP^ PK is un-
altered, § 139. Hence the elements IH^ KL still attract /'equally,
and the proposition follows as in § 479.
Hence inside the shell the potential is constant.
537. Two confocal shells (§ 535) being given, the potential of the
first at any point, P^ of the surface of the second, is to that of the
second at the corresponding point,/, on the surface of the first, as
the mass of the first is to the mass of the second. This beautiful
proposition is due to Chasles.
To any element of the mass of the outer shell at Q corresponds an
element of mass of the inner at q, and these bear the same ratio to
the whole masses of their respective shells, that the corresponding
element of the spherical shell from which either may be derived bears
to its whole mass. Whence, since Pq = 0^, the proposition is true
for the corresponding elements at Q and ^, and therefore for the
entire shells.
Also, as the potential of a shell on an internal point is constant,
and as one of two confocal ellipsoids is wholly within the other : it
follows that the external equipotential surfaces for any such shell are
confocal ellipsoids, and therefore that the attraction of the shell on an
external point is normal to a confocal ellipsoid passing through the
point.
538. Now it has been shown (§ 495) that the attraction of a shell
on an external point near its surface exceeds that on an internal
point infinitely near it by 47rp where p is the surface-density of the
shell at that point. Hence, as (§ 536) there is no attraction on an
internal point, the attraction of a shell on a point at its exterior
surface is 47rp: or 47rp/ if p be now put for the volume-density, and t
for the (infinitely small) thickness of the shell, § 495. From this
it is easy to obtain by integration the determination of the whole
attraction of a homogeneous ellipsoid on an external particle.
13 — 2
196 ABSTRACT DYNAMICS.
539. The following splendid theorem is due to Maclaurin : —
The attractions exerted by two homogeneous and confocal ellipsoids
on the same poifit external to each, or external to one and on the sur-
face of the other, are in the same directioft and proportional to their
masses.
540. Ivory's theorem is as follows : —
Let correspondtJig points F, p, be taken on the surfaces of two homo-
geneous co?ifocal ellipsoids, E, e. The x compo7ient of the attraction of E
on p, is to that of e on P as the area of the section of E by the plane of
yz is to that of the coplanar section of e.
Poisson showed that this theorem is true for any law of force
whatever. This is easily proved by employing in the general ex-
pressions for the components of the attraction of any body, after one
integration, the properties of corresponding points upon confocal
ellipsoids (§ 535).
541. An ingenious application of Ivory's theorem, by Duhamel,
must not be omitted here. Concentric spheres are a particular case
of confocal ellipsoids, and therefore the attraction of any sphere on
a point on the surface of an internal concentric sphere, is to that of
the latter upon a point in the surface of the former as the squares of
the radii of the spheres. Now if the law of attraction be such that a
homogeneous spherical shell of tmiforjfi thick7iess exerts 7io attraction on
a7i inter7ial poi7tt, the action of the larger sphere on the internal point
is reducible to that of the smaller. Hence the law is that of the i7i-
verse square of the dista7ice, as is easily seen by making the smaller
sphere less and less till it becomes a mere particle. This theorem is
due originally to Cavendish.
542. {Def7uti07z.) If the action of terrestrial or other gravity on
a rigid body is reducible to a single force in a line passing always
through one point fixed relatively to the body, whatever be its position
relatively to the earth or other attracting mass, that point is called its
centre of gravity, and the body is called a ce7itrobaric body.
543. One of the most startling result-s of Green's wonderful theory
of the potential is its establishment of the existence of centrobaric
bodies; and the discovery of their properties is not the least curious
and interesting among its very various applications.
544. If a body {B) is centrobaric relatively to any one attracting
mass {A), it is centrobaric relatively to every other: and it attracts
all matter external to itself as if its own mass were collected in its
centre of gravity.'
545. Hence §§ 510, 515 show that —
{a) The ce7itre of gravity of a ce7itrobaric body necessarily lies in its
interior; or in other words, ca7t only be reached from external space by
a path cutting through some of its 77iass. And
{b) No centrobaric body can co7isist ofpa7'ts isolated frofn one another,
^ Thomson. Proc. R.S.E., Feb. 1864.
STATICS OF A PARTICLE.— ATTRACTION, 197
each in space externa! to all: in other words, the outer boundary of every
centroharic body is a si7igle closed surface.
Thus we see, by {a)^ that no symmetrical ring, or hollow cylinder
with open ends, can have a centre of gravity; for its centre of
gravity, if it had one, would be in its axis, and therefore external to
its mass.
546. If any 77tass whatever^ M^ and any single surface, *S, com-
pletely enclosing it be given, a distributioJi of any given amount, M\
of matter on this surface may be found which shall make the whole
centrobaric with its centre of gravity in any given position (G) within
that surface.
The condition here to be fulfilled is to distribute M' over S, so as
by it to produce the potential
AI^-M' _
EG
at any point, E, of S\ V denoting the potential of M at this point.
The possibiUty and singleness of the solution of this problem were
stated above (§ 517). It is to be remarked, however, that if M' be
not given in sufficient amount, an extra quantity must be taken, but
neutralized by an equal quantity of negative matter, to constitute the
required distribution on S.
The case in which there is no given body M to begin with is
important; and yields the following: —
547. A given quantity of matter may he distributed in one way, but
in only one way, over any given closed surface, so as to constitute a
centrobaric body with its centre of gravity at any given point within it.
Thus we have already seen that the condition is fulfilled by making
the density inversely as the distance from the given point, if the
surface be spherical. From what was proved in §§ 519, 524 above,
it appears also that a centrobaric shell may be made of either half of
the lemniscate in the diagram of § 526, or of any of the ovals within
it, by distributing matter with density proportional to the resultant
force oim 2X I and ;;/ at /' ; and that the one of these points which
is within it is its centre of gravity. And generally, by drawing the
equipotential surfaces relatively to a mass m collected at a point /,
and any other distribution of matter whatever not surrounding this
point ; and by taking one of these surfaces which encloses / but
no other part of the mass, we learn, by Green's general theorem,
and the special proposition of § 524, how to distribute matter
over it so as to make it a centrobaric shell with / for centre of
gravity.
548. Under hydrokinetics the same problem will be solved for a
cube, or a rectangular parallelepiped in general, in terms of con-
verging series; and under electricity (in a subsequent volume) it will
be solved in finite algebraic terms for the surface of a lens bounded
by two spherical surfaces cutting one another at any sub-multiple of
two right angles, and for either part obtained by dividing this surface
198 ABSTRACT DYNAMICS.
in two by a third spherical surface cutting each of its sides at right
angles.
549. Matter may be distributed in an iftfi?iite number of ways
throughout a given closed space, to constitute a centrobaric body with its
centre of gravity at any given poi?it within it.
For by an infinite number of surfaces, each enclosing the given
point, the whole space between this point and the given closed surface
may be divided into infinitely thin shells; and matter may be dis-
tributed on each of these so as to make it centrobaric with its centre
of gravity at the given point. Both the forms of these shells and the
quantities of matter distributed on them, may be arbitrarily varied in
an infinite variety of ways.
Thus, for example, if the given closed surface be the pointed oval
constituted by either half of the lemniscate of the diagram of § 526,
and if the given point be the point / within it, a centrobaric solid
may be built up of the interior ovals with matter distributed over
them to make them centrobaric shells as above (§ 547). From what
was proved in § 534, we see that a solid sphere with its density
varying inversely as the fifth power of the distance from an external
point, is centrobaric, and that its centre of gravity is the image (§ 530)
of this point relatively to its surface.
550. The centre of gravity of a centrobaric body composed of
true gravitating matter is its centre of inertia. For a centrobaric
body, if attracted only by another infinitely distant body, or by matter
so distributed round itself as to produce (§ 517) uniform force in
parallel lines throughout the space occupied by it, experiences (§ 544)
a resultant force always through its centre of gravity. But in this
case this force is the resultant of parallel forces on all the particles of
the body, which (see Properties of Matter, below) are rigorously pro-
portional to their masses: and it is proved that the resultant of
such a system of parallel forces passes through the point defined in
§ 195, as the centre of inertia.
551. The moments of inertia of a centrobaric body are equal
round all axes through its centre of inertia. In other words (§ 239),
all these axes are principal axes, and the body is kinetically sym-
metrical round its centre of inertia.
CHAPTER VII.
STATICS OF SOLIDS AND FLUIDS.
552. Forces whose lines meet. Let ABC be a rigid body acted
on by two forces, F and (2, applied to it
at different points, D and E respectively, in
lines in the same plane.
Since the lines are not parallel, they will
meet if produced; let them be produced and
meet in O. Transmit the forces to act on
that point; and the result is that we have
simply the case of two forces acting on a
material point, which has been already con-
sidered.
553. The preceding solution is applicable to every case of non-
parallel forces in a plane, however far removed the point may be in
which their Hnes of action meet, and the resultant will of course be
found by the parallelogram of forces. The limiting case of parallel
forces, or forces whose lines of action, however far produced, do not
meet, was considered above, and the position and magnitude of
the resultant were investigated. The following is an independent
demonstration of the conclusion arrived at.
554. Parallel forces in a plane. The resultant of two parallel
forces is equal to their sum, and is in the parallel line which divides
any line drawn across their Hnes of action into parts inversely as their
magnitudes.
1°. Let P and Q be two parallel forces acting on a rigid body in
similar directions in lines AB and CD. Draw any line AC across
their lines. In it introduce any
pair of balancing forces. Sin AG
and 5 in CH. These forces will
not disturb the equilibrium of
the body. Suppose the forces
F and S'm. AG, and Q and S in
CH^ to act respectively on the
points A and C of the rigid body.
The forces F and S, in AB and
AGy have a single resultant in
some line AM, within the angle
200 ABSTRACT DYNAMICS.
GAB; and Q and S in CD and CH have a resultant in some line
CN, within the angle DCff.
The angles MAC, NCA are together greater than two right .
angles, hence the lines MA^ NC will meet if produced. Let them
meet in O. Now the two forces P and S may be transferred to
parallel lines through O. Similarly the forces g and 5 may be also
transferred. Then there are four forces acting on (9, two of which,
S in OK and 6* in OL^ are equal and directly opposed. They may,
therefore, be removed, and there are left two forces equal to P and
(2 in one line on O^ which are equivalent to a single force P-\- Q'ln
the same line.
2°. If, for a moment, we suppose OE to represent the force P,
then the force representing S must be equal and parallel to EA^ since
the resultant of the two is in the direction OA. That is to say,
S\P'.\EA\ 0E\
and in like manner, by considering the forces S in OL and Q in OE,
we find that
Q'.SwOE'.EC.
Compounding these analogies, we get at once
Q: P'.'.EA'.ECy
that is, the parts into which the line is divided by the resultant are
inversely as the forces.
555. Forces in dissimilar directions. The resultant of two parallel
forces in dissimilar directions^, of which one is greater than the other,
is found by the following rule : Draw any line across the lines of the
forces and produce it across the line of the greater, until the whole
line is to the part produced as the greater force is to the less ; a force
equal to the excess of the greater force above the less, applied at the
extremity of this line in a parallel line and in the direction similar to
that of the greater, is the resultant of the system.
Let /'and Q in KK' and LL\ be the
contrary forces. From any point A^ in
• the line of P, draw a line AB across the
line of Q cutting it in B, and produce the
-p line to E, so that AE : BE :: Q \ P.
Through E draw a line MM' parallel to
K li iW KK' or LL'.
In MM' introduce a pair of balancing forces each equal to Q- P.
Then P in AK' and Q~ P in EM have a resultant equal to their
1 In future the word 'contrary' will be employed instead of the phrase
'parallel and in dissimilar directions' to designate merely directional opposiiiotty
while the unqualified word 'opposite' will be understood to signify contrary and
in one line.
STATICS OF SOLIDS AND FLUIDS. 201
sum, or Q. This resultant is in the line LL' j for, from the ana-
logy*
AEvBEv. Q:F,
we have AE-BE \ BE w Q- P '. F,
or AB :BEy. Q~F\F.
Hence F in AK\ Q in BL', and Q-F in EM are in equilibrium
and may be removed. There remains only Q-Fm EM\ which is
therefore the resultant of the two given forces. This fails when the
forces are equal
556. Any number of parallel forces in a plane. Let F^, F^, F^,
etc., be any number of parallel forces
acting on a rigid body in one plane. j I I I /
To find their resultant in position and ^ L^ If^'L^ L^ U^
magnitude, draw any line across their r»J f 1 i if
lines of action, cutting them in points, ^J \pjfa -?[ ^f
denoted respectively by ^ J, ^2, ^3, etc.,
and in it choose a point of reference O. Let the distances of the
lines of the forces from this point be denoted by a^, a^^ ^3, etc.; as
OA^ = a^, OA^ = a^, etc. Also let F denote the resultant, and x its
distance from O.
Find the resultant of any two of the forces, as F^ and F^, by
§ 554. Then if we denote this resultant by F', we have
F'^F, + F^.
Divide A^ A^ in E' into parts inversely as the forces, so that
F,xA,E' = F^y<E'A^.
Hence if we denote OE' by x' we have
F,x{x'-a,) = F^x{a^-x')
or (F, + F^)x'==F,a, + F^a^,
that is F' x' = F^a^-¥ F^ a^ .
Similarly we shall find the resultant of R' and F^ to- be
F'^R'^F, = F,^F,^F,',
and R"x" = R'xf + F^a^^F^a^+F^a^ + F^a^.
Hence, finally we have
R^F,^F,^F^^ + /'„ (i),
and Rx=^F^a^-\-F^a^ + F^a^+ ■.+^„^,. •••(2).
In this method negative forces or negative values of any of the
quantities a^,a^, ..., maybe included, provided the generalized rules of
multiplication and addition in algebra are followed.
557. Any number of parallel forces not in one plane. To find the
resultant, let a plane cut the lines of all the forces, and let the points
0
n
N,
K
J^'JV^
202 ABSTRACT DYNAMICS.
in which they are cut be specified by reference to two rectangular
axes in the plane. Let the plane be YOX\ OX^ OV, the axes of
reference, O the origin of co-ordinates, and A^, A^, A^, etc., the points
in which the plane cuts the lines of the forces, 1*^^, T^, T^, etc. Thus
each of these points will be specified by perpendiculars drawn from
it to the axis. Let the co-ordinates
of the point A^ be denoted by x^ yyy
of A^, by x^ y^; and so on; that is,
OJV, = x„ N, A, =y,; OJV^ = x^, N^ A^
=y^, etc.; let also the final resultant be
^j denoted by R, and its co-ordinates by
X and y.
Find the resultant of /\ and P^ by
joining Aj^ A„, and dividing the line
-jV^-^ inversely as the forces. Suppose E'
the point in which this resultant cuts
the plane of reference. Then
T,xA,E' = F,xE^A,.
To find the co-ordinates, which may be denoted by x'y', of the
point E' with reference to OX and O V; draw E'JV' perpendicular
to OX and cutting it in JV', and from A^ draw A^ K parallel to OX,
or perpendicular to A^N^, and cutting it in ^ and E'N' in M.
Then (Euclid vi. 2)
A,E' :E'A^::A,M:MX.
Hence T^ x A,M= P^ x MK,
or P^{x'-x:)--P^Kx^-x'\
whence we get {P^ + P^ x' = P^x^ -1- P^x^ ;
and since P' = P^ + P^^
we have P'x' = P^x^ + P^x^ ,
and similarly, P'y' = P^y^ -{- P^y^.
We may find the resultant of P' and P^ in like manner, and so
with all the forces. Hence we have for the final resultant,
ji==A+p,+j',+ +j>„..
Py = P,y, + P,y, + P,j>,+ ... +P,y,
(3),
(4),
(s).
These equations may include negative forces, or negative co-
ordinates.
558. Conditions of equilibrium of any number of parallel forces.
In order that any given parallel forces may be in equilibrium, it is
not sufificient alone, that their algebraic sum be equal to zero.
For, let P^P^ + P^ + etc. - o.
STATICS OF SOLIDS AND FLUIDS. 203
From this equation it follows that if the forces be divided into two
groups, one consisting of the forces reckoned positive, the other of
those reckoned negative, the sum, or resultant (§ 556), of the former
is equal to the resultant of the latter; that is, \i ^R and 'F denote the
resultants of the positive and negative groups respectively,
But unless these resultants are directly opposed they do not balance
one another; wherefore, if {x^y) and (x'y) be the co-ordinates of
^F and 'F respectively, we must have for equilibrium
and ,y = 'y',
whence we get ,F ^x - 'F'x = o
and ,Fj-'Fy = o.
But ^F ^x is equal to the sum of those of the terms Fy_x^, F^^, etc.,
which are positive, and 'F'x is equal to the sum of the others each
with its sign changed : and so for ^F^y and 'F'y. Hence the pre-
ceding equations are equivalent to
F^x^-¥ F^^+ +Fjx:^^=^o.
■^lJl + -^2j2+ +F^j^=o.
We conclude that, for equilibrium, it is necessary and sufficient that
each of the following three equations be satisfied : —
F, + F, + F^+ +F„=o (6),
F^x^ + Fjc^ + F^x.^ + + F^x^ = o (7),
F,y,+F,y, + F,y,+ +^„y. = o (8).
559. If equation (6) do not hold, but equations (7) and (8) do, the
forces have a single resultant through the origin of co-ordinates. If
equation (6) and either of the other two do not hold, there will be a
single resultant in a hne through the corresponding axis of reference,
the co-ordinates of the other vanishing. If equation (6) and either of
the other two do hold, the system is reducible to a single couple in a
plane through that hne of reference for which the sum of the products
is not equal to nothing. If the plane of reference is perpendicular to
the lines of the forces, the moment of this couple is equal to the sum
of the products not equal to nothing.
560. In finding the resultant of two contrary forces in any case in
which the forces are unequal — the smaller the difference of magnitude
between them, the farther removed is the point of application of the
resultant. When the difference is nothing, the point is removed to
an infinite distance, and the construction (§ 555) is thus rendered
nugatory. The general solution gives in this case F = 0; yet the
forces are not in equilibrium, since they are not directly opposed.
Hence two equal contrary forces neither balance, nor have a single
resultant. It is clear that they have a tendency to turn the body to
204 ABSTRA CT D YNAMICS.
which they are applied. This system was by Poinsot denominated
a couple.
In actual cases the direction of a couple is generally reckoned
positive if the couple tends to turn contrary to the hands of a watch
as seen by a person looking at its face, negative when it tends to
turn with the hands. Hence the axis, which may be taken to repre-
sent a couple, will show, if drawn according to the rule given in § 201,
whether the couple is positive or negative, according to the side of its
plane from which it is regarded.
561. Proposition I. Any two couples in the same or in parallel
planes are in equilibrium if their moments are equal and they tend to
turn in contrary directions.
1°. Let the forces of the first couple be parallel to those of the
second, and let all four forces be in one plane.
n T A^ n^ -^^^ ^^ forces of the first couple be
o xr A o ^ .^ ^^ ^^^ ^j^^ ^^^ ^^ ^^^ second
F' in A'B' and CD'. Draw any line
EF' across the lines of the forces, cut-
ting them respectively in points F, Fy
E' and F' 'y then the moment of the
B BR B JD' first couple IS Z'. ^i^ and of the second
F', E'F' ; and since the moments are equal we have
F.EF=F'.E'F'.
Of the four forces, P m AB and P' in CFf act in similar direc-
tions, and F in CD and F' in A'B' also act in similar directions;
and their resultants respectively can be determined by the general
method (§ 556). The resultant of F in AB, and F' in CD', is thus
found to be equal to /*+ F'^ and if HL is the line in which it acts,
F.EK=F'.KF'.
Again, we have F. EF= F' , E'F'.
Subtract the first member of the latter equation from the first
member of the former, and the second member of the latter from the
second member of the former : there remains
F.FK=F'.KE',
from which we conclude, that the resultant of F in CD and F' in
A'B' is in the line LH. Its magnitude is /*+ F', Thus the given
system is reduced to two equal resultants acting in opposite directions
in the same straight line. These balance one another, and therefore
the given system is in equilibrium.
Corollary. A couple may be transferred from its own arm to any
other arm in the same line, if its moment be not altered.
562. Proposition I. 2". All four forces in one plane, but those of
one couple not parallel to those of the other.
Produce their lines to meet in four points; and consider the paral-
lelogram thus formed. The products of the sides, each into its per-
pendicular distance from the side parallel to it, are equal, each product
STATICS OF SOLIDS AND FLUIDS. 205
being the area of the parallelogram. Hence, since the moments of the
two couples are equal, their forces are proportional to the sides of the
parallelogram along which they act. And, since the couples tend to
turn in opposite directions, the four forces represented by the sides of
a parallelogram act in similar directions relatively to the angles, and
dissimilar directions in the parallels, and therefore balance one
another.
Corollary. The statical effect of a couple is not altered, if its arm
be turned round any point in the plane of the couple.
563. Proposition I. 3^ The two couples not in the same plane,
but the forces equal and parallel.
Let there be two couples, acting re- •M' pp,
spectively on arms EF and E'F\ which E^-^rrr^^ — L"*" ^ ..F'
are parallel but not in the same plane. .-------r-C.''* *^ |p/
Join EF' and E'F. These lines bisect ^r"^ |^p7"''-^
one another in O. \p
Of the four forces, F on F and F' on
E' act in similar directions, and their resultant, equal io F+ F\ may
be substituted for them. It acts in a parallel line through O. Simi-
larly F on E and F' on F'' have also a resultant equal to F+ F'
through O; but these resultants being equal and opposite, balance,
and therefore the given system is in equilibrium.
Remark i. — A corresponding demonstration may be appHed to
every case of two couples, the moments of which are equal, though
the forces and arms may be unequal. When the forces and arms are
unequal, the lines EF'j E'F cut one another in O into parts inversely
as the forces.
Remark 2. — Hence as an extreme case, Proposition I, 1°, may be
brought under this head. Let EF be the arm of one couple, EF'
of the other, both in one straight line. Join FE' , and divide it
inversely as the forces. Then FK : KE' w EF \ E'F' and EF' is
divided in the same ratio.
Corollary. Transposition of couples. Any two couples in the
same or in parallel planes, are equivalent, provided their moments
are equal, and they tend to turn in similar directions.
564. Proposition IL Any number of couples in the same or in
parallel planes, may be reduced to a single
resultant couple, whose moment is equal to the L
algebraic sum of their moments, and whose
plane is parallel to their planes.
Reduce all the couples to forces acting on
one arm AB^ which may be denoted by a. A^
Then if /\, F^, P,, etc., be the forces, the mo- yi\
ments of the couples will be F^a^ F^a, F^a, ,,^
etc. Thus we have /\, F^, F^, etc., in AK, p
reducible to a single force, their sum, and ^^
similarly, a single force F^ + F^-^ etc., in FL.
2o6 ABSTRACT DYNAMICS.
These two forces constitute a couple whose moment is (/\ + P.
+ i^3 + etc.) a. But this product is equal to P^a-\- P^a + P^a + etc.,
the sum of the moments of the given couples, and therefore any
number of couples, etc. If any of the couples act in the direction
opposite to that reckoned positive, their moments must be reckoned
as negative in the sum.
565. Proposition III. Any two couples not in parallel planes
may be reduced to a single resultant couple, whose axis is the
diagonal through the point of reference of the parallelogram de-
scribed upon their axes.
1°. Let the planes of the two couples cut the plane of the diagram
perpendicularly in the lines AA' and
BB' respectively; let the planes of
the couples also cut each other in a
line cutting the plane of the diagram
in O. Through O, as a point of re-
ference, draw OK the axis of the first
couple, and OL the axis of the se-
cond. On OK and OL construct the
^ ^ parallelogram OKML. Its diagonal
" OM is the axis of the resultant couple.
Let the moment of the couple acting in the plane BB\ be denoted
by G, and of that in AA\ by H. For the given couples, substitute
two others, with arms equal respectively to G and H\, and therefore
with forces equal to unity.
From OB and OA measure off OE = G, and 0F= H, and let these
lines be taken as the arms of the two couples respectively. The
forces of the couples will thus be perpendicular to the plane of the
diagram : those of the first, acting outwards at E^ and inwards at O \
and those of the second, outwards at (7, and inwards at F. Thus, of
the four equal forces which we have in all, there are two equal and
opposite at (9, which therefore balance one another, and may be
removed; and there remain two equal parallel forces, one acting
outwards at E, and the other inwards at F^ which constitute a couple
on an arm EF.
This single couple is therefore equivalent to the two given couples.
2°. It remains to be proved that its axis is OM. Join EF, As,
by construction, OL and OK are respectively perpendicular to OA^
and OB, the angle KOL is equal to the angle A OB'. Hence, MLO
the supplement of the former is equal to EOF, the supplement of the
latter. But OK is equal to OE ; each being equal to the moment of
the first of the given couples ; and therefore LM, which is equal to
the former, is equal to OE. Similarly OL is equal to OF. Thus there
are two triangles, MLO and EOF, with two sides of one respectively
equal to two sides of the other, and the contained angles equal : there-
fore the remaining sides OM, EF are equal, and the angles LOM\
OFE are equal. But since OL is perpendicular to OF, OM is
STATICS OF SOLIDS AND FLUIDS. 207
perpendicular to EF. Hence OM is the axis of the resultant
couple.
566. Proposition IV. Any number of couples whatever are either
in equilibrium with one another, or may be reduced to a single couple,
under precisely the same conditions as those already investigated for
forces acting on one point, the axes of the couples being now taken
everywhere instead of the lines formerly used to represent the forces.
1°. Resolve each couple into three components having their axes
along three rectangular lines of reference, OX^ O F, OZ. Add all
the components corresponding to each of these three lines. Then
if the resultant of all the couples whose axes are along the line
OX^ be denoted by Z,
OY, „ „ M,
OZ, „ „ iV^
and if G be the resultant of these three, we have
G^ J{L' + M' + N'):
and if ^, -q, 0, be the angles which the axis of this couple G, makes
with the three axes OX, O V, OZ, respectively, we have
. Z M.N
cos 4= 7,; CO^f]=-^\ cos^ = ^.
Lr Lr Lr
567. 2°. Conditions of equilibrium of any number of couples. For
equilibrium the resultant couple must be equal to nothing : but as it is
compounded of three subsidiary resultant couples in planes at right
angles to one another, they also must each be equal to nothing. The
remarks already made, and the equations already given in §§ 471, 472,
apply with the necessary modification to couples also. Thus, for
instance, the equations of equilibrium are
G^ cos ^j + G^ cos ^3 + G.^ cos t,^ + etc. = o,
G^ cosr7j + G^ cos 173 + 6^3 cos t]^ + etc. = o,
C?i cos ^1 + 6^2 cos ^2 + G^ cos ^3 + etc. ^ o.
568. Before investigating the conditions of equilibrium of any
number of forces acting on a rigid body, we shall establish some
preliminary propositions.
1°. A force and a couple in the same or in parallel planes may be
reduced to a single force. Let the plane of the couple be the plane
of the diagram, and let its moment be
denoted by G. Let R, acting in the
line OA in the same plane, be the force.
Transfer the couple to an arm (which may
be denoted by a) through the point O, such
that each force shall be equal to R; and let
its position be so chosen, that one of the
forces shall act in the same straight line with
R in OA, but in the opposite direction to it.
G
208
ABSTRACT DYNAMICS.
R and G being known, the length of this arm can be found, for
since the moment of the transposed couple is
Ra^G
G
we have
a-
R'
Through O then, draw a line OCf perpendicular to OA^ making it
equal to a. On this arm apply the couple, a force equal to R^ acting
on O' in a line perpendicular to 00\ and another in the opposite
direction at the other extremity. There are now three forces, two of
which, being equal and opposite to one another, in the line AA\ may
be removed. One, acting on the point 0\ remains, which is there-
fore equivalent to the given system.
569. 2°. A couple and a force in a given line inclined to its plane
may be reduced to a smaller couple in a plane perpendicular to the
force, and a force equal and parallel to the given force.
Let OA be the line of action of the force /?,
and let OK be the axis of the couple. Let
the moment be denoted by G : and let A OK,
the inclination of its axis to the line of the
force, be 0. Draw OB perpendicular to OA.
By Prop. IV. (§ 566) resolve the couple into
two components, one acting round OA as
axis, and one round OB. Thus the compo-
nent round OA will be
G cos B,
and the component round OB^
G sin Q.
Now as G sin B acts in the same plane as the given force -^, this com-
ponent together with R may be reduced by § 568 to one force. This
force which is equal to R, will act not at O in the line OA^ but in a
parallel line through a point O' out of the plane of the diagram. Thus
the given system is reduced to a smaller couple G cos ^, and to a
force in a line which, by Poinsot, was denominated the central axis of
the system.
670. 3'
a couple.
Let /*, acting
Any number of forces may be reduced to a force and
on J/j be one of a number of forces acting in
different directions on different points of a rigid
body. Choose any point of reference O, for the
different forces, and through it draw a line AA'
parallel to the line of the first force P^. Through (?,
draw 00' perpendicular to A A or the line of the
force P^. In the line A A' introduce two equal op-
posite forces, each equal to P^. There are now
three forces, producing the same effect as the given
force, and they may be grouped differently : P^ acting
STATICS OF SOLIDS AND FLUIDS. 209
in O in the line OA^ and a couple, Py^ acting at O' , and P^ at O in
the line 0A\ on an arm 00'. Reduce similarly all. the other forces,
each to a force acting on (9, and to a couple. But all the couples
thus obtained are equivalent to a single couple, and all the forces are
equivalent to one force. Hence, &c.
571. Reduction of any number of forces to their simplest equi-
valent system.
Suppose any number of forces acting in any directions on different
points of a rigid body. Choose three rectangular planes of reference
meeting in a point (9, the origin of co-ordinates. In order to effect
the reduction it is necessary to bring in all the forces to the point O.
This may be done in two different ways — either in two steps, or
directly.
572. 1°. Let the magnitudes of the forces be jPi, P^, &c., and the
co-ordinates, with reference to the rectangular planes, of the points at
which they act respectively, be (^1, Ji, 2J, {x^., y^, z.^, &c. Let also
the direction cosines be (Z^, m^, n^, (4, m,, wj, &c. Resolve each
force into three components, parallel to OX^ O V, OZ, respectively.
Thus, if (Xj, Vi, Z,), &c., be the components of /\, &c., we shall
have
X, = P,/,; X,= PJ,; &c. (1)
V,=^P,m,; V,^P„m,; &c. (2)
Z, = P,n,;Z, = P,n,',&:c. (3)
To transfer these components to the point O. Let ^i, in AfJ^, be
the component, parallel to OX, of the force P^^ acting on the point Af.
From M transmit it along its line to a
point JV in the plane ZO Y: the co-or-
dinates of this point will bejj, z^. From
iVdraw a perpendicular NB to OY,
and through B draw a line parallel to
MK or OX. Introducing in this line
a pair of balancing forces each equal
to Xj, we have a couple acting on an ^^ q
arm z^ in a plane parallel to XOZ, '^'"^y^ 27
and a single force X^ parallel to OX Y'^
in the plane XOY. The moment of
this couple is X^z^, and its axis is along
OY. Next transfer the force X^ from B to O, by introducing a pair
of balancing forces in X' OX, one of which, with the force X^ in the
line through B parallel to XX and the direction similar to OX, form
a couple acting on an arm y^ . This couple, wKen y^ and X^ are both
positive, tends to turn in the plane XOY from O Fto OX. Therefore
by the rule, § 201, its axis must be drawn from O in the direction 0Z\
Hence its moment is to be reckoned as- X^y^. Besides this couple
there remains a single force equal to X^, in the direction OX, through
the point O. Similarly by successive steps transfer the forces Y^, Z^,
T. 14
2IO ABSTRACT DYNAMICS.
to the origin of co-ordinates. In this way six couples of transference
are got, three tending to turn in one direction round the axes respec-
tively, and three in the opposite direction; and three single forces at
right angles to one another, acting at the point O. Thus for the force
P^, at the point (^i, Ji, z^, we have as equivalent to it at the point
O, three forces X^, Y^, Z^, and three couples;
Zj^ -Y^z^\ moment of the couple round 0X\ (4)
X^z^-Z^x^\ moment of the couple round OY \ (5)
Y^x^-X^y^', moment of the couple round OZ. (6)
All the forces may be brought in to the origin of co-ordinates in a
similar way.
573. 2°. Otherwise: Let P be one of the forces acting in the line
MT on a point Moi a rigid body. Let
O be the origin of co-ordinates; OX,
OY, OZ, three rectangular lines of re-
ference. Join OMa-nd produce the line
to S. From O draw OJV, cutting at
right angles in the point JV, the line
MT produced through Af. Let OJV
be denoted by/, and the angle TMS
by K. In a line through O parallel to
MT (not shown in diagram) suppose
introduced a pair of balancing forces each equal to P. We have
thus a single force equal to P acting at O, and a couple, whose
moment is Pj>, in the plane ONM. The direction cosines of this
plane, or, which is the same thing, the direction cosines of a per-
pendicular to it, that is, the axis of the couple are (§ 464), if we denote
them by <^, Xj '/'> respectively,
y z
-n--m
sm K
z , X
-I-- n
r r
r r
sm K
Now in the triangle ONM^
ON^ OMsm OMN,
that is / = r sin k.
STATICS OF SOLIDS AND FLUIDS. 211
Hence, if we substitute / for its value in the three preceding equations,
the expression for the direction cosines are reduced to
(7)
(8)
p (9)
To find the component couples round OX, O Y, OZ, multiply these
direction cosines respectively by Fj>', whence we get
Fp .f^=^F{7iy - mz), moment of couple round OX, ( i o)
Fp .x-F{lz- 7tx), moment of couple round OY, (11)
Fp.\^ — F {mx - ly)j moment of couple round OZ. (12)
That this result is the same as that got by the other method will be
evident, by considering that (equations i, 2, 3),
Fl=X- Fm=.Y', Fn = Z.
574. When by either of the methods all the forces have been re-
ferred to 6>, there is obtained a set of couples acting round OX, O Y,
OZ; and a set of forces acting along OX, O Y, OZ. Find then the
resultant moments of all the couples ; and the sums of all the forces :
if L, M, N be the resultant moments round OX, O Y, OZ respectively,
we have
Z = (Z, y, - Y^ z,) + (4^, - F, z^) + &c. (13)
M= (Xj z^ -Z^ x^ + (Xj z^ -Z2 x^ + &c. (14)
N= ( y; X, -X,y,) +( y, x, -X,y,) + &c. (15)
and if X, Y, Z be the resultant forces,
X=X,+X, + X^ + 8zc. (i6>
Y=Y,+ Y,+ Y, + &c. (17).
Z=Z, + Z^+ Z^ + &LC. (18)
575. Finally, find the resultant of the three forces by the formulae
of Chap. VI, and the resultant of the three couples by Prop. IV
(§ 566). Thus, if /, m, n be the direction cosines of the resultant force
i?, we have (§§ 463, 467)
, X Y Z , .
and if X, fi, v be the direction cosines of the axis of the resultant
couple, we have (§ 566)
^ L M N . .
X=g; /x = ^; v=^. (20)
14—2
2 1 2 ABSTRA CT D YNAMICS.
676. Conditions of Equilibrium. The conditions of equilibrium
of three forces at right angles to one another have been already stated
in § 470; and the conditions for three rectangular couples in § 567.
If a body be acted on by three forces and three couples simul-
taneously, all the conditions applicable when they act separately, must
also be satisfied when they act conjointly, since a force cannot balance
a couple. Six Equations of Equilibrium therefore are necessary and
sufficient for a rigid body acted on by any number of forces. These
are
/*! cos a^ + P^ cos a^ + &c. = o,
P^ cos ft + P^ cos ft + &c. = o,
P^ cos y^ + P^ cos 72 + &c. = o,
G^ cos 4 + G^ cos ^3 + &c. = o,
G^ cos T7i + G^ cos 772 + &c. = o,
G^ cos ^1 + G^ cos ^a + &c. = o.
577. If the line of the resultant found by § 575, is perpendicular
to the plane of the couple, that is, if
X = /, iL-m^ v=n;
X=Y = z^ (21)
the system cannot be reduced to another with a force and a smaller
couple, and in this case the line found for the resultant force is the
central axis of the system.
578. If, on the other hand, the plane of the couple is parallel to the
line of the force, or the axis of the couple perpendicular to the line of
the force, that is, if
/A. + mix + nv = Oj
or ZX + My+JVZ=o, (22)
the force and couple may (§ 568) be reduced to one force : and this
G
force is parallel to the former, at a distance from it equal to -^ , in
the plane of it and the couple. Thus, X(7 being the foot of the
perpendicular from the origin on the line of action of the resultant
force, 0(7 wiW be perpendicular to the Hne of the resultant force, and
to the axis of the resultant couple, and therefore its direction cosines
are (§464, ^);
mv — /I IX, nX — Ivj //x - m\, (23)
each of which will be positive when O' lies within the solid angle
Q
edged by OX^ OY, OZ. Hence, remembering that 00' = -^^ and
using the expressions (19) and (20), we find for the co-ordinates of (7
YN-ZM ZL-XN XM-YL
7?' ' ^» ' R^
(24)
STATICS OF SOLIDS AND FLUIDS. 213
and we thus complete the specification of the single force to which
the system is reduced when (22) holds.
679. If the line of the force is inclined at any angle to the plane
of the couple, the resultant system can be further reduced by § 569,
to a smaller couple and a force in a determinate line, the * central
axis.' This couple is G cos 6, and according to the notation, may be
thus expressed by § 464, (7), if we substitute the values given in (19)
and (20),
. XL+YM+ZN , .
Gcose = . (25)
The other component couple, G sin 9, lies in the same plane as R,
and with it may be reduced by §568 to one force, which will be
parallel to R, that is, in the direction (/, m, n)^ at a distance from it
equal to — ^ — . Hence the direction cosines of 00' will be
mv — n\k n\ — lv l}x. — m\ , r\
sin (9 ' sin^ ' sin ^ * ' ^ '
Substituting in each of these for /, X, &c., their respective values,
and multiplying each member by — ^ — , we have for the co-ordinates
of the point 6>', as in § 578,
YN-ZM ZL-XN XM-YL . .
R' ' R' ' R' ' ^^'^
A single force, R, through the point thus specified in the direction
(/, My n)j with a couple in a plane perpendicular to it, and having .
XL + YM+ZN
R
for its moment, is consequently the system oi force along central axis
and mifitmum couple, to which the given set of forces is determinately
reducible by Poinsot's beautiful method.
580. The position of the central axis may be determined other-
wise j thus, instead of in the first place bringing the forces to O, bring
them to any point T, of which let (x, y, z) be the co-ordinates. Then
instead of Y^z^+Y^z^ + ^c, which we had before (§ 574), we have now
Y,{z,-z)+Y,{z,-z) + &ic.,
or Y^ z^ + Y^z^ -f &c. - ( y; + Fa -f- &c.) z,
and so for the others. Then for the moments of the couples of trans-
ference we have
a = Z -{Zy-Yz\
0i = M-(Xz-Zx),
M = N-(Yx^Xy).
Now, let T be chosen, if possible, so as to make the resultant
214 ABSTRACT DYNAMICS.
eouple lie in a plane perpendicular to it. The condition to be ful-
filled in this case is
X Y Z'
which, when for 3£, &c., we substitute their values, becomes,
L-{Zy-Yz) _ M-{Xz-Zx) _ N-(Yx-Xy)
X ~ Y ~ Z '
which is the equation of the central axis of the system.
To show that O', the point determined in §§ 578, 579, is in the
central axis thus found ; we have, substituting for ^, y, z, the values
given in (24),
Z {ZL - XN) + Y (XM- YL)
1 =^-
Reducing, and remarking that
LR'-LY'-LZ' = LX\
we find that the first member becomes
X
and is therefore equal to each of the two others. Thus is verified the
comparison of the two methods.
581. In one respect, this reduction of a system of forces to a
couple, and a force perpendicular to its plane, is the best and simplest,
especially in having the advantage of being determinate, and it gives
very clear and useful conceptions regarding the effect of force on
a rigid body. The system may, however, be farther reduced to two
equal forces acting symmetrically on the rigid body, but whose po-
sition is indeterminate. Thus, supposing the central axis of the
system has been found, draw a line AA', at right angles through any
point C in it, so that CA may be equal to CA\ For R, acting along
the central axis, substitute \R at each end of A A'. Thus, choosing
this line A A' as the arm of the couple, and calling it a, we have at
each extremity of it two forces, — perpendicular to the central axis,
and ^R parallel to the central axis. Compounding these, we get
two forces, each equal to (\R^-\--y) , through A and A' re-
spectively, perpendicular to AA\ and equally inclined at the angle
tan~^ — p on the two sides of the plane through A A' and the central
axis.
582. It is obvious, from the formulae of § 195, that if masses pro-
portional to the forces be placed at the several points of application
of these forces, the centre of inertia of these masses will be the same
STATICS OF SOLIDS AND FLUIDS. 215.
point in the body as the centre of parallel forces. Hence the re-
actions of the different parts of a rigid body against acceleration in
parallel lines are rigorously reducible to one force, acting at the centre
of inertia. The same is true approximately of the action of gravity
on a rigid body of small dimensions relatively to the earth, and hence
the centre of inertia is sometimes (§ 195) called the Centre of Gravity.
But, except on a centrobaric body (§ 543), gravity is not in general
reducible to a single force : and when it is so, this force does not pass
through a point fixed relatively to the body in all positions.
583. The resultant of a system of parallel forces is not a single
force when the algebraic sum of the given forces vanishes. In
this case the resultant is a couple whose plane is parallel to the
common direction of the forces. A good example of this is furnished
by a magnetized mass, of steel, of moderate dimensions, subject to the
influence of the earth's magnetism only. As will be shown later, the
amounts of the so-called north and south magnetisms in each element
of the mass are equal, and are therefore subject to equal and opposite
forces, all parallel to the line of dip. Thus a compass-needle expe-
riences from the earth's magnetism merely a couple or directive action,
and is not attracted or repelled as a whole.
584. If three forces, acting on a rigid body, produce equilibrium,
their directions must lie in one plane ; and must all meet in one point,
or be parallel. For the proof, we may introduce a consideration
which will be very useful to us in investigations connected with the
statics of flexible bodies and fluids.
If ajiy forces^ acting on a solid or fluid body, produce equilibrium, we
may suppose any portions of the body to becoffie fixed, or rigid, or rigid
aftd fixed, without destroying the equilibrium.
Applying this principle to the case above, suppose any two points
of the body, respectively in the lines of action of two of the forces, to
be fixed — the third force must have no moment along the line joining
these points; that is, its direction must pass through the line joining
them. As any two points in the lines of action may be taken, it
follows that the three forces are coplanar. And three forces in one
plane cannot equilibrate, unless their- directions are parallel or pass
through a point.
585. It is easy and useful to consider various cases of equilibrium
when no forces act on a rigid body but gravity and the pressures,
normal or tangential, between it and fixed supports. Thus, if one
given point only of the body be fixed, it is evident that the centre of
gravity must be in the vertical line through this point — else the weight
and the reaction of the support would form an unbalanced couple.
Also for stable equilibrium the centre of gravity must be below the
point of suspension. Thus a body of any form may be made to
stand in stable equilibrium on the point of a needle if we rigidly
attach to it such a mass as to cause the joint centre of gravity to be
below the point of the needle.
2l6
ABSTRACT DYNAMICS.
586. An interesting case of equilibrium is suggested by what are
called Rocking Stones, where, whether by natural or by artificial pro-
cesses, the lower surface of a loose mass of rock is worn into a convex
form which may be approximately spherical, while the bed of rock on
which it rests in equilibrium is, whether convex or concave, also ap-
proximately spherical, if not plane. A loaded sphere resting on a
spherical surface is therefore a type of such cases.
Let O, O' be the centres of curvature of the fixed and rocking
bodies respectively, when in the position of equilibrium.
Take any two infinitely small equal arcs FQ, Pp ; and
at Q make the angle O'QR equal to FOp. When, by
displacement, Q and p become the points in contact,
QR will evidently be vertical ; and, if the centre of
gravity G, which must be in OPO' when the movable
body is in its position of equilibrium, be to the left of
QR, the equilibrium will obviously be stable. Hence,
if it be below i?, the equilibrium is stable, and not
unless.
Now if p and o- be the radii of curvature OF, O' P
of the two surfaces, and 6 the angle POp, the angle
f\
QO'R will be equal to — ; and we have in the triangle
(?<^'^(§ii9)
Hence
i?6>':cr::sin^:sin('^-f-^^
: : o- : cr + p (approximately).
PR = <T-
cr + p
and therefore, for stable equilibrium.
p + a
PG<
pa_
p + 0-
If the lower surface be plane, p is infinite, and the condition becomes
(as in § 256)
PG<(T.
If the lower surface be concave, the sign of p must be changed, and
the condition becomes
PG
pa-
which cannot be negative, since p must be numerically greater than o-
in this case.
587. If two points be fixed, the only motion of which the system is
capable is one of rotation about a fixed axis. The centre of gravity
must then be in the vertical plane passing through those points, and
below the line adjoining them for stable equilibrium.
588. If a rigid body rest on a fixed surface, there will in general be
only three points of contact, § 380 ; and the body will be in stable
STATICS OF SOLIDS AND FLUIDS 217
equilibrium if the vertical line drawn from its centre of gravity cuts
the plane of these three points within the triangle of which they form
the corners. For if one of these supports be removed, the body will
obviously tend to fall towards that support. Hence each of the three
prevents the body from rotating about the line joining the other two.
Thus, for instance, a body stands stably on an inclined plane (if the
friction be sufficient to prevent it from sliding down) when the vertical
line drawn through its centre of gravity falls within the base, or area
bounded by the shortest line which can be drawn round the portion in
contact with the plane. Hence a body, which cannot stand on a
horizontal plane, may stand on an inclined plane.
589. A curious theorem, due to Pappus, but commonly attributed
to Guldinus, may be mentioned here, as it is employed with advantage
in some cases in finding the centre of gravity of a body — though it is
really one of the geometrical properties of the Centre of Inertia. It is
obvious from § 195. If a plane dosed curve revolve through any a7igle
about an axis in its plane ^ the solid content of the surface gefierated is
equal to the product of the area of either end into the length of the path
described by its centre of gravity ; and the area of the curved surface is
equal to the product of the length of the curve into the length of the path
described by its centre of gravity.
590. The general principles upon which forces of constraint and
friction are to be treated have been stated above (§§ 258, 405). We
add here a few examples, for the sake of illustrating the application
of these principles to the equilibrium of a rigid body in some of the
more important practical cases of constraint.
591. The application of statical principles to the Mechanical Powers^
or elementary machines, and to their combinations, however complex,
requires merely a statement of their kinematical relations (as in §§ 91,
97, 113, &c.) and an immediate translation into Dynamics by New-
ton's principle (§241); or by Lagrange's Virtual Velocities (§ 254),
with special attention to the introduction of forces of friction, as in
§ 405. In no case can this process involve further difficulties than
are implied in seeking the geometrical circumstances of any infinitely
small disturbance, and in the subsequent solution of the equations to
which the translation into dynamics leads us. We will not, therefore,
stop to discuss any of these questions ; but will take a few examples
of no very great difficulty, before for a time quitting this part of the
subject. The principles already developed will be of constant use to
us in the remainder of the work, which will furnish us with ever-
recurring opportunities of exemplifying their use and mode of appli-
cation.
Let us begin with the case of the Balance, of which we promised
(§ 384) to give an investigation.
592. Ex. 1. We will assume the line joining the points of attach-
ment of the scale-pans to the arms to be at right angles to the line
joining the centre of gravity of the beam with the fulcrum. It is
2i8 ABSTRACT DYNAMICS,
obvious that the centre of gravity of the beam must not coincide with
the knife-edge, else the beam would rest indifferently in any position.
We will suppose, in the first place, that the arms are not of equal length.
Let O be the fulcrum, G the
centre of gravity of the beam,
M its mass ; and suppose that
with loads P and Q, in the pans
the beam rests (as drawn) in a
position making an angle B with
the horizontal line.
Taking moments about (?,
and, for convenience (see § 185),
using gravitation measurement of the forces, we have
Q {AB cos e+OA sin 6) + M. OG sin 6 = F{AC cos 6 - OA sin 0).
From this we find
P.AC-Q.AB
{F+Q)OA + M.OG'
If the arms be equal we have
tang (i--0^^
{F+Q) OA + M.OG'
Hence the Sensibility (§ 384) is greater, (i) as the arms are longer,
(2) as the mass of the beam is less, (3) as the fulcrum is nearer to the
line joining the points of attachment of the pans, (4) as the fulcrum is
nearer to the centre of gravity of the beam. If the fulcrum be in the
line joining the points of attachment of the pans, the sensibility is the
same for the same difference of loads in the pan.
To determine the Stability we must investigate the time of oscilla-
tion of the balance when slightly disturbed. It will be seen, by refer-
ence to a future chapter, that the equation of motion is approximately
{Mk' + {F+Q) OB'} e + Qg (AB cos 6 + OA sin 0)
+ MgOG sin d - Fg {AC cos $ - OA sin 6) = o,
^ being the radius of gyration (§ 235) of the beam. If we suppose
the arms and their loads equal, we have for the time of an infinitely
small oscillation
7;
M/e'+2F.0B'
{2F.0A+M.0G)g'
Thus the stability is greater for a given load, (i) the less the length of
the beam, (2) the less its mass, (3) the less its radius of gyration, (4)
the further the fulcrum from the beam, and from its centre of gravity.
With the exception of the second, these adjustments are the very
opposite of those required for sensibility. Hence all we can do is to
effect a judicious compromise; but the less the mass of the beam, the
better will the balance be, in ^of/i respects.
The general equation, above written, shows that if the length, and
the radius of gyration, of one arm be diminished, the corresponding
STATICS OF SOLIDS AND FLUIDS.
219
load being increased so as to maintain equilibrium — a form of balance
occasionally useful — the sensibility is increased.
Fx. II. Find the position of equilibrium of a rod AB resting on a
smooth horizontal rail D, its lower end pressing against a smooth
vertical wall A C parallel to the rail.
The figure represents a vertical section through the rod, which
must evidently be in a plane perpendicular to the wall and rail.
The only forces acting are three, R the pressure of the wall on the
rod, horizontal; S that of the
rail on the rod, perpendicular
to the rod; W the weight of
the rod, acting vertically down-
wards at its centre of gravity.
If the half-length of the rod be
a, and the distance of the rail
from the wall b^ these are given
— and all that is wanted to fix
the position of equilibrium is the angle the rod makes with the wall.
Call CAB, e. Then we see at once that AD = -r— 7: .
sm 0
Resolving horizontally R- S cos ^ = o, . ( i )
vertically W- *Ssin ^ = o. (2)
Taking moments about A,
S.AD- JVa sin 0*0,
or Sb-JVasm'0 = o. (3)
As there are only three unknown quantities, R, S, and 0, these three
equations contain the complete solution of the problem. By (2)
and (3)
sin'^ = -, which gives 0.
W
Hence by (2) S=- — ;-,
^ ^ ' sm ^ '
and by (i) R = Scos 6 = Wcot 0.
Fx. III. As an additional example, suppose the wall and rail to be
rough, and /x to be the co-efficient of statical friction for both. If the
rod be placed in the position of equilibrium just investigated for the
case of no friction, none will be called into play, for there will be no
tendency to motion to be overcome. If the end A be brought lower
and lower, more and more friction will be called into play to over-
come the tendency of the rod to fall between the wall and the rail,
until we come to a limiting position in which motion is about to
commence. At that instant the friction at y^ is /x, times the pressure
ABSTRACT DYNAMICS,
on the wall, and acts upwards. That at Z> is /* times the pressure
on the rod, and acts in the direction DB. CaUing CAD = 0^ in this
case, our three equations become
B^ + IxS^smO^- S^cosO^ =o, (ii)
IV-fXjR^ — Si sin 0^ - fxS^ cos ^^ = o, (21)
S.d-Wasm'e^ =0. (3i)
The directions of both the friction-forces passing through A, neither
appears in (31). This is why A is preferable to any other point about
which to take moments.
By eliminating B^ and S^ from these equations we get
I - - sm^ 0^ = fJL - sin^ 6^ {2 cosO^-fx sin ^1), (4 )
from which 0^ is to be found. Then S^ is known from (31), and B^
from either of the others.
If the end A be raised above the position of equilibrium without
friction, the tendency is for the rod to fall outside the rail ; more and
more friction will be called into play, till the position of the rod (^2)
is such that the friction reaches its greatest value, /x times the pressure.
We may thus find another limiting "position for stability; and between
these the rod is in equilibrium in any position.
It is useful to observe that in this second case the direction of each
friction is the opposite to that in the former, and the same equations
will serve for both if we adopt the analytical artifice of changing the
sign of [J.. Thus for 0^, by (41),
I -Tsin^^2 =
fjL T sin^^2 (2 cos O^ + fi sin 6^.
M
Ex. IV. A rectangular block lies on a rough horizontal plane, and
is acted on by a horizontal force whose line of action is midway be-
tween two of the ver-
tical sides. Find the
magnitude of the force
when just sufficient to
produce motion, and
whether the motion will
be of the nature of slid-
ing or overturning.
If the force B tends to
overturn the body, it is
evident that it will turn about the edge A, and therefore the pressure, B^
of the plane and the friction, 5, act at that edge. Our statical condi-
tions are, of course B= JV
S=B
Wb = Ba
where b is half the length of the solid, and a the distance of P from
the plane. From these we have 5= - W.
a
c
\
D
A
P
G
\
t
,
(
A
STATICS OF SOLIDS AND FLUIDS.
221
Now ^S* cannot exceed ijlR, whence we must not have - greater
than /A, if it is to be possible to upset the body by a horizontal force
in the line given for F.
. A simple geometrical construction enables us to solve this and similar
problems, and will be seen at once to be merely a graphic representa-
tion of the above process. Thus if we produce the directions of
the applied force, and of the weight, to meet in ZT, and make at A the
angle BAK whose co-tangent is the co-efficient of friction : there will
be a tendency to upset, or not, according as ZTis above, or below, AK.
Ex. V. A mass, such as a gate, is supported by two rings, A and
B, which pass loosely round a rough
vertical post. In equilibrium, it is ob-
vious that at A the part of the ring
nearest the mass, and at B the farthest
from it, will be in contact with the post.
The pressures exerted on the rings, R
and 6", will evidently have the directions
AC^ CB, indicated in the diagram. If
no other force besides gravity act on the
mass, the line of action of its weight, IV,
must pass through the point C (§ 584).
And it is obvious that, however small be
the co-efficient of friction, provided there
be friction at all, equilibrium is always
possible if the distance of the centre of gravity from the post be great
enough compared with the distance between the rings.
When the mass is just about to slide down, the full amount of
friction is called into play, and the angles which R and .S make with
the horizon are each equal to the angle of repose. If we draw A C,
BC according to this condition, then for equilibrium the centre of
gravity G must not lie between the post and the vertical line through
the point C thus determined. If, as in the figure, G lies in the ver-
tical line through C, then a force applied upwards at Q^, or down-
wards at Q^, will remove the tendency to fall; but a force applied
upwards at Q^, or downwards at Q^, will produce sliding at once.
A similar investigation is easily applied to the jamming of a sliding
piece or drawer, and to the determination of the proper point of appli-
cation of a force to move it. This we leave to the student.
As an illustration of the use of friction, let us consider a cord
wound round a rough cylinder, and on the point of sHding.
Neglecting the weight of the cord, which is small in practice com-
pared with the other forces; and con-
sidering a small portion AB of the
cord, such that the tangents at its
extremities include a very small angle
6] let T' be the tension at one end,
22 2 ABSTRACT DYNAMICS.
T' at the other, p the pressure of the rope on the cylinder per unit
of length.
Then/.^^ = 2rsin -= 2"^ approximately. Also \y.p.AB= T - T
when the rope is just about to slip, i.e.
fx,Te=r-T,
or r={i+fie)T.
Hence, for equal small deflections, 0, of the rope, the tension
increases in the geometrical ratio (i + fx6) : i ; and thus by a common
theorem (compound interest payable every instant) we have T= £'**7^,
if T, T^ be the tensions at the ends of a cord wrapped on a cylinder,
when the external angle between the directions of the free [ends is a.
[c is the base of Napier's Logarithms.] We thus obtain the singular
result, that the dimensions of the cylinder have no influence on the
increase of tension by friction, provided the cord is perfectly flexible.
593. Having thus briefly considered the equilibrium of a rigid
body, we propose, before entering upon the subject of deformation
of elastic solids, to consider certain intermediate cases, in each of
which a particular assumption is made the basis of the investiga-
tion— thereby avoiding a very considerable amount of analytical
difficulties.
594. Very excellent examples of this kind are furnished by the
statics of a flexible and inextensible cord or chain, fixed at both ends,
and subject to the action of any forces. The curve in which the
chain hangs in any case may be called a Catenary, although the term
is usually restricted to the case of a uniform chain acted on by gravity
only.
595. We may consider separately the conditions of equilibrium of
each element; or we may apply the general condition (§ 257) that the
whole potential energy is a minimum, in the case of any conservative
system of forces; or, especially when gravity is the only external
force, we may consider the equilibrium of a 7f;2//<? portion of the chain
treated for the time as a rigid body (§ 584).
596. The first of these methods gives immediately the three follow-
ing equations of equilibrium, for the catenary in general : —
(i) The rate of variation of the tension per unit of length along
the cord is equal to the tangential component of the applied force,
per unit of length.
(2) The plane of curvature of the cord contains the normal com-
ponent of the applied force, and the centre of curvature is on the
opposite side of the arc from that towards which this force acts.
(3) The amount of the curvature is equal to the normal component
of the applied force per unit of length at any point divided by the ten-
sion of the cord at the same point.
The first of these is simply the equation of equilibrium of an
infinitely small element of the cord relatively to tangential motion.
The second and third express that the component of the resultant
STATICS OF SOLIDS AND FLUIDS. 223
of the tensions at the two ends of an infinitely small arc, along the
normal through its middle point, is directly opposed and is equal to
the normal applied force, and is equal to the whole amount of it on
the arc. For the plane of the tangent Hnes in which those tensions
act is (§ 12) the plane of curvature. And if ^ be the angle between
them (or the infinitely small angle by which the angle between their
positive directions falls short of tt), and T the arithmetical mean of
their magnitudes, the component of their resultant along the line
bisecting the angle between their positive directions is 2 Z'sin ^0, rigor-
ously: or TO, since 6 is infinitely small. Hence T0=^N6s if hs be
the length of the arc, and N^s the whole amount of normal force
applied to it. But (§ 9) ^ = — if p be the radius of curvature; and
therefore —= -^.
P T
which is the equation stated in words (3) above.
597. From (1) of § 596, we see that if the appHed forces on any
particle of the cord constitute a conservative system, and if any equal
infinitely small lengths of the string experience the same force and
in the same direction when brought into any one position by motion
of the string, the difference of the tensions of the cord at any two
points of it when hanging in equilibrium, is equal to the difference
of the potential (§ 504) of the forces between the positions occupied
by these points. Hence, whatever the position where the potential is
reckoned zero, the tension of the string at any point is equal to the
potential at the position occupied by it, with a constant added.
598. From § 596 it follows immediately that if a material particle
of unit mass be carried along any catenary with a velocity, s, equal
to T, the numerical measure of the tension at any point, the force
upon it by which this is done is in the same direction as the resultant
of the applied force on the catenary at this point, and is equal to
the amount of this force per unit of length, multiplied by T. For
denoting by S the tangential, and (as before) by N the normal
component of the applied force per unit of length at any point P
of the catenary, we have, by § 596 (i), 6" for the rate of variation of
s per unit length, and therefore Ss for its variation per unit of time.
That is to say, 's^Ss^ ST,
or (§ 225) the tangential component force on the moving particle
is equal to ST. Again, by § 596 (3),
T^ i'
NT=^- = -,
P P
or the centrifugal force of the moving particle in the circle of cur-
vature of its path, that is to say, the normal component of the
force on it, is equal to JVT. And lastly, by (2) this force is in
the same direction as JV. We see therefore that the direction of the
224 ABSTRACT DYNAMICS.
whole force on the moving particle is the same as that of the
resultant of 6" and N) and its magnitude is T times the magnitude
of this resultant.
599. Thus we see how, from the more familiar problems of the
kinetics of a particle, we may immediately derive curious cases
of catenaries. For instance : a particle under the influence of a
constant force in parallel lines moves in a parabola with its axis
vertical, with velocity at each point equal to that generated by
the force acting through a space equal to its distance from the
directrix. Hence, if z denote this distance, and / the constant
force, T= J 2fz
in the allied parabolic catenary; and the force on the catenary is
parallel to the axis, and is equal in amount per unit of length, to
J2fz V 2^
Hence if the force on the catenary be that of gravity, it must have
its axis vertical (its vertex downwards of course for stable equili-
brium) and its mass per unit length at any point must be inversely
as the square root of the distance of this point above the directrix.
From this it follows that the whole weight of any arc of it is
proportional to its horizontal projection.
600. Or, if the question be, to find what force towards a given
fixed point, will cause a cord to hang in any given plane curve with
this point in its plane; it may be answered immediately from the
solution of the coresponding problem in 'central forces.'
601. When a perfectly flexible string is stretched over a smooth
surface, and acted on by no other force throughout its length than
the resistance of this surface, it will, when in stable equilibrium,
lie along a line of minimum length on the surface, between any
two of its points. For (§ 584) its equilibrium can be neither
disturbed nor rendered unstable by placing staples over it, through
which it is free to slip, at any two points where it rests on the
surface: and for the intermediate part the energy criterion of stable
equilibrium is that just stated.
There being no tangential force on the string in this case, and the
normal force upon it being along the normal to the surface, its oscu-
lating plane (§ 596) must cut the surface everywhere at right angles.
These considerations, easily translated into pure geometry, establish
the fundamental property of the geodetic lines on any surface. The
analytical investigations of the question, when adapted to the case
of a chain of not given length, stretched between two given points on
a given smooth surface, constitute the direct analytical demonstration
of this property.
In this case it is obvious that the tension of the string is the same
at every point, and the pressure of the surface upon it is [§ 596 (3)]
at each point proportional to the curvature of the string.
STATICS OF SOLIDS AND FLUIDS. 225
602. No real surface being perfectly smooth, a cord or chain may
rest upon it when stretched over so great a length of a geodetic on a
convex rigid body as to be not of minimum length between its
extreme points : but practically, as in tying a cord round a ball,
for permanent security it is necessary, by staples or otherwise, to
constrain it from lateral slipping at successive points near enough
to one another to make each free portion a true minimum . on the
surface.
603. A very important practical case is supplied by the con-
sideration of a rope wound round a rough cylinder. We may
suppose it to lie in a plane perpendicular to the axis, as we thus
simplify the question very considerably without sensibly injuring
the utility of the solution. To simplify still further, we shall suppose
that no forces act on the rope but tensions and the reaction of
the cylinder. In practice this is equivalent to the supposition that
the tensions and reactions are very large compared with the weight
of the rope or chain ; which, however, is inadmissible in some
important cases, especially such as occur in the application of the
principle to brakes for laying submarine cables, to ergometers, and
to windlasses (or capstans with horizontal axes).
By § 592 we have r- T^cm®,
showing that, for equal successive amounts of integral curvature
(§ 14), the tension of the rope augments in geometrical progression.
To give an idea of the magnitudes involved, suppose /u, = -5, ^ = tt, then
r = r.c -5^ = 4 . 8 1 7; roughly.
Hence if the rope be wound three times round the post or cylinder
the ratio of the tensions of its ends, when motion is about to com-
mence, is
5" : I or about 15,000 : i.
Thus we see how, by the aid of friction, one man may easily check
the motion of the largest vessel, by the simple expedient of coiling a
rope a few times round a post. This application of friction is
of great importance in many other applications, especially to ergome-
ters (§§ 389, 390).
604. With the aid of the preceding investigations, the student
may easily work out for himself the solution of the general problem
of a cord under the action of any forces, and constrained by a
rough surface; it is not of sufficient importance or interest to find
a place here.
605. An elongated body of elastic material, which for brevity
we shall generally call a wire, bent or twisted to any degree, sub-
ject only to the condition that the radius of curvature and the reci-
procal of the twist are everywhere very great in comparison with
the greatest transverse dimension, presents a case in which, as we
T. 15
2 26 ABSTRACT DYNAMICS.
shall see, the solution of the general equations for the equilibrium
of an elastic solid is either obtainable in finite terms, or is reducible
to comparatively easy questions agreeing in mathematical conditions
with some of the most elementary problems of hydrokinetics, elec-
tricity, and thermal conduction. And it is only for the determination
of certain constants depending on the section of the wire and the
elastic quality of its substance, which measure its flexural and
torsional rigidity, that the solutions of these problems are required.
When the constants of flexure and torsion are known, as we shall
now suppose them to be, whether from theoretical calculation or
experiment, the investigation of the form and twist of any length
of the wire, under the influence of any forces which do not produce
a violation of the condition stated above, becomes a subject of
mathematical analysis involving only such principles and fornmlae
as those that constitute the theory of curvature (§§ 9-15) and twist
in geometry or kinematics.
606. Before entering on the general theory of elastic solids, we
shall therefore, according to the plan proposed in § 593, examine
the dynamic properties and investigate the conditions of equilibrium
of a perfectly elastic wire, without admitting any other condition or
limitation of the circumstances than what is stated in § 605, and
without assuming any special quality of isotropy, or of crystalline,
fibrous or laminated structure in the substance.
607. Besides showing how the constants of flexural and tor-
sional rigidity are to be determined theoretically from the form of
the transverse section of the wire, and the proper data as to the
elastic qualities of its substance, the complete theory simply in-
dicates that, provided the conditional limit of deformation is not
exceeded, the following laws will be obeyed by the wire under
stress : —
Let the whole mutual action between the parts of the wire on the
two sides of the cross section at any point (being of course the action
of the matter infinitely near this plane on one side, upon the matter
infinitely near it on the other side), be reduced to a single force
through any point of the section and a single couple. Then —
I. The twist and curvature of the wire in the neighbourhood of
this section are independent of the force, and depend solely on the
couple.
II. The curvatures and rates of twist producible by any several
couples separately, constitute, if geometrically compounded, the curva-
ture and rate of twist which are actually produced by a mutual action
equal to the resultant of those couples.
608. It may be added, although not necessary for our present
purpose, that there is one determinate point in the cross section
such that if it be chosen as the point to which the forces are trans-
ferred, a higher order of approximation is obtained for the fulfilment
STATICS OF SOLID S AND FLUIDS 227
of these laws than if any other point of the section be taken. That
point, which in the case of a wire of substance uniform through its
cross section is the centre of inertia of the area of the section, we
shall generally call the elastic centre, or the centre of elasticity, of
the section. It has also the following important property : — The line
of elastic centres, or, as we shall call it, the elastic central line, remains
sensibly unchanged in length to whatever stress within our conditional
limits (§ 605) the wire be subjected. The elongation or contraction
produced by the neglected resultant force, if this is in such a direction
as to produce any, will cause the line of rigorously no elongation to
deviate only infinitesimally from the elastic central line, in any part
of the wire finitely curved. It will, however, clearly cause there to
be no line of rigorously unchanged lengthy in any straight part of the
wire : but as the whole elongation would be infinitesimal in com-
parison with the effective actions with which we are concerned, this
case constitutes no exception to the preceding statement.
609. In the most important practical cases, as we shall see later,
those namely in which the substance is either ' isotropic,' which is
sensibly the case with common metallic wires, or has an axis of
elastic symmetry along the length of the piece, one of the three
normal axes of torsion and flexure coincides with the length of the
wire, and the two others are perpendicular to it ; the first being an
axis of pure torsion, and the two others axes of pure flexure. Thus
opposing couples round the axis of the wire twist it simply without
bending it ; and opposing couples in either of the two principal planes
of flexure, bend it into a circle.
610. In the more particular case in which two principal rigidities
against flexure are equal, every plane through the length of the wire
is a principal plane of flexure, and the rigidity against flexure is equal
in all. This is clearly the case with a common round wire, or rod, or
with one of square section. It can be shown to be the case for a
rod of isotropic material and of any form of normal section which
is ' kinetically symmetrical' (§ 239) round all axes in its plane through
its centre of inertia.
611. In this case, if one end of the rod or wire be held fixed, and
a couple be applied in any plane to the other end, a uniform spiral
form will be produced round an axis perpendicular to the plane
of the couple. The lines of the substance parallel to the axis of
the spiral are not, however, parallel to their original positions: and
lines traced along the surface of the wire parallel to its length when
straight, become as it were secondary spirals, circling round the
main spiral formed by the central line of the deformed wire. Lastly,
in the present case, if we suppose the normal section of the wire
to be circular, and trace uniform spirals along its surface when
deformed in the manner supposed (two of which, for instance, are
the lines along which it is touched by the inscribed and the circum-
15—2
2 28 ABSTRACT DYNAMICS,
scribed cylinder), these lines do not become straight, but become
spirals laid on as it were round the wire, when it is allowed to take
its natural straight and untwisted condition.
612. A wire of equal flexibility in all directions may clearly be held
in any specified spiral form, and twisted to any stated degree, by a
determinate force and couple applied at one end, the other end being
held fixed. The direction of the force must be parallel to the axis
of the spiral, and, with the couple, must constitute a system of which
this line is (§ 579) the central axis : since otherwise there could not
be the same system of balancing forces in every normal section of
the spiral. All this may be seen clearly by supposing the wire to
be first brought by any means to the specified condition of strain ;
then to have rigid planes rigidly attached to its two ends perpendicular
to its axis, and these planes to be rigidly connected by a bar lying
in this line. The spiral wire now left to itself cannot but be in
equilibrium : although if it be too long (according to its form and
degree of twist) the equilibrium may be unstable. The force along
the central axis, and the couple, are to be determined by the condition
that, when the force is transferred after Poinsot's manner to the elastic
centre of any normal section, they give two couples together equiva-
lent to the elastic couples of flexure and torsion.
613. A wire of equal flexibility in all directions may be held in
any stated spiral form by a simple force along its axis between rigid
pieces rigidly attached to its two ends, provided that, along with its
spiral form a certain degree of twist be given to it. The force is
determined by the condition that its moment round the perpendicular
through any point of the spiral to its osculating plane at that point,
must be equal and opposite to the elastic unbending couple. The
degree of twist is that due (by the simple equation of torsion) to the
moment of the- force thus determined, round the tangent at any point
of the spiral. The direction of the force being, according to the
preceding condition, such as to press together the ends of the spiral,
the direction of the twist in the wire is opposite to that of the tortuosity
(§ 13) of its central curve.
614. The principles with which we have just been occupied are
immediately applicable to the theory of spiral springs; and we
shall therefore make a short digression on this curious and im-
portant practical subject before completing our investigation of elastic
curves.
A common spiral spring consists of a uniform wire shaped per-
manently to have, when unstrained, the form of a regular helix, with
the principal axes of flexure and torsion everywhere similarly situated
relatively to the curve. When used in the proper manner, it is
acted on, through arms or plates rigidly attached to its ends, by forces
such that its form as altered by them is still a regular helix. This
condition is obviously fulfilled if (one terminal being held fixed) an
STATICS OF SOLIDS AND FLUIDS. 229
infinitely small force and infinitely small eouple be applied to the
other terminal along the axis, and in a plane perpendicular to it, and
if the force and couple be increased to any degree, and always kept
along and in the plane perpendicular to the axis of the altered spiral.
It would, however, introduce useless complication to work out the
details of the problem except for the case (§ 609) in which one of
the principal axes coincides with the tangent to the central line, and
is therefore an axis of pure torsion, as spiral springs in practice
always belong to this case. On the other hand, a very interesting
complication occurs if we suppose (what is easily realized in practice,
though to be avoided if merely a good spring is desired) the normal
section of the wire to be of such a figure, and so situated relatively
to the spiral, that the planes of greatest and least flexural rigidity
are obHque to the tangent plane of the cylinder. Such a spring when
acted on in the regular manner at its ends must experience a certain
degree of turning through its whole length round its elastic central
curve in order that the flexural couple developed may be, as we
shall immediately see it must be, precisely in the osculating plane of
the altered spiral. All that is interesting in this very curious effect is
illustrated later in full detail (§ 624 of our larger work) in the case of an
open circular arc altered by a couple in its own plane, into a circular
arc of greater or less radius ; and for brevity and simplicity we shall
confine the detailed investigation of spiral springs on which we now
enter, to the cases in which either the wire is of equal flexural rigidity
in all directions, or the two principal planes of (greatest and least or
least and greatest) flexural rigidity coincide respectively with the
tangent plane to the cylinder, and the normal plane touching the
central curve of the wire, at any point.
615. The axial force, on the movable terminal of the spring, trans-
ferred according to Poinsot to any point in the elastic central
curve, gives a couple in the plane through that point and the axis
of the spiral. The resultant of this and the couple which we suppose
applied to the terminal in the plane perpendicular to the axis of the
spiral is the effective bending and twisting couple : and as it is in a
plane perpendicular to the tangent plane to the cylinder, the com-
ponent of it to which bending is due must be also perpendicular to
this plane, and therefore is in the osculating plane of the spiral. This
component couple therefore simply maintains a curvature different
from the natural curvature of the wire, and the other, that is, the
couple in the plane normal to the central curve, pure torsion.
The equations of equilibrium merely express this in mathematical
language.
616. The potential energy of the strained spring is
l[^(t^-^/ + ^TT/,
if A denote the torsional rigidity, B the flexural rigidity in the plane
of curvature, w and tzr^ the strained and unstrained curvatures, and t
230 ABSTRACT DYNAMICS.
the torsion of the wire in the strained condition, the torsion being
reckoned as zero in the unstrained condition. The axial force, and
the couple, required to hold the spring to any given length reckoned
along the axis of the spiral, and to any given angle between planes
through its ends and the axes, are of course (§ 244) equal to the rates
of variation of the potential energy, per unit of variation of these
co-ordinates respectively. It must be carefully remarked, hovv^ever,
that, if the terminal rigidly attached to one end of the spring be held
fast, so as to fix the tangent at this end, and the motion of the other
terminal be so regulated as to keep the figure of the intermediate
spring always truly spiral, this motion will be somewhat complicated ;
as the radius of the cylinder, the inclination of the axis of the spiral
to the fixed direction of the tangent at the fixed end, and the position
of the point in the axis in which it is cut by the plane perpendicular
to it through the fixed end of the spring, all vary as the spring changes
in figure. The effective components of any infinitely small motion of
the movable terminal are its component translation along, and rota-
tion round, the instantaneous position of the axis of the spiral [two
degrees of freedom], along with which it will generally have an
infinitely small translation in some direction and rotation round some
line, each perpendicular to this axis, and determined from the two
degrees of arbitrary motion, by the condition that the curve remains a
true spiral.
617. In the practical use of spiral springs, this condition is not
rigorously fulfilled : but, instead, one of two plans is generally fol-
lowed: — (i) Force, without any couple, is applied pulling out or
pressing together two definite points of the two terminals, each as
nearly as may be in the axis of the unstrained spiral; or (2) One
terminal being held fixed, the other is allowed to slide, without any
turning, in a fixed direction, being as nearly as may be the direction
of the axis of the spiral when unstrained. The preceding investiga-
tion is applicable to the infinitely small displacement in either case :
the couple being put equal to zero for case (i), and the instantaneous
rotatory motion round the axis of the spiral equal to zero for
case (2).
618. In a spiral spring of infinitely small inclination to the plane
perpendicular to its axis, the displacement produced in the movable
terminal by a force applied to it in the axis of the spiral is a simple
rectilineal translation in the direction of the axis, and is equal to the
length of the circular arc through which an equal force carries one
end of a rigid arm or crank equal in length to the radius of the
cylinder, attached perpendicularly to one end of the wire of the spring
supposed straightened and held with the other end absolutely fixed,
and the end which bears the crank, free to turn in a collar. This
statement is due to J. Thomson \ who showed that in pulling out
a spiral spring of infinitely small incHnation the action exercised and
1 Camb. ami Dub. Math. Jour. 1848.
STATICS OF SOLIDS AND FLUIDS. 231
the elastic quality used are the same as in a torsion-balance with the
same wire straightened (§ 386). This theory is, as he proved ex-
perimentally, sufficiently approximate for most practical applications ;
spiral springs, as commonly made and used, being of very small
inclination. There is no difficulty in finding the requisite correction,
for the actual inclination in any case. The fundamental principle that
spiral springs act chiefly by torsion seems to have been first discovered
by Binet in i8i4\
619. Returning to the case of a uniform wire straight and untwisted
(that is, cylindrical or prismatic) when free from stress; let us suppose
one end to be held fixed in a given direction, and no other force
from without to influence it except that of a rigid frame attached to
its other end acted on by a force, i?, in a given line, AB, and a
couple, G, in a plane perpendicular to this line. The form and twist
it will have when in equilibrium are determined by the condition that
the torsion and flexure at any point, F, of its length are those due to
the couple G compounded with the couple obtained by bringing R
to F.
620. Kirchhoflf has made a very remarkable comparison between
the static problem of bending and twisting a wire, and the kinetic
problem of the rotation of a rigid body. We can give here
but one instance, the simplest of all — the Elastic Curve of James
Bernoulli, and the common pendulum. A uniform straight wire,
either equally flexible in all planes through its length, or having its
directions of maximum and minimum flexural rigidity in two planes
through its whole length, is acted on by a force and couple in one
of these planes, applied either directly to one end, or by means of an
arm rigidly attached to it, the other end being held fast. The force
and couple may, of course (§ 568), be reduced to a single force, the
extreme case of a couple being mathematically included as an in-
finitely small force at an infinitely great distance. To avoid any
restriction of the problem, we must suppose this force applied to an
arm rigidly attached to the wire, although in any case in which the
line of the force cuts the wire, the force may be applied directly at
the point of intersection, without altering the circumstances of the
wire between this point and the fixed end. The wire will, in these
circumstances, be bent into a curve lying throughout in the plane
through its fixed end and the line of the force, and (§ 609) its curva-
tures at difl"erent points will, as was first shown by James Bernoulli,
be simply as their distances from this line. The curve fulfilling this
condition has clearly just two independent parameters, of which one
is conveniently regarded as the mean proportional, a, between the
radius of curvature at any point and its distance from the line of force,
and the other, the maximum distance, b, of the wire from the line of
force. By choosing any value for each of these parameters it is easy
to trace the corresponding curve with a very high approximation to
accuracy, by commencing with a small circular arc touching at one
^ St Venant, Coviptes Rcndtts, Sept. 1864.
232 ABSTRACT DYNAMICS.
extremity a straight line at the given maximum distance from the line
of force, and continuing by small circular arcs, with the proper
increasing radii, according to the diminishing distances of their middle
points from the line of force. The annexed diagrams are, however,
not so drawn, but are simply traced from the forms actually assumed
by a flat steel spring, of small enough breadth not to be much
disturbed by tortuosity in the cases in which different parts of it cross
one another. The mode of application of the force is sufficiently
explained by the indications in the diagram.
621. As we choose particularly the common pendulum for the
corresponding kinetic problem, the force acting on the rigid body in
the comparison must be that of gravity in the vertical through its
centre of gravity. It is convenient, accordingly, not to take imity as
the velocity for the point of comparison along the bent wire, but the
velocity which gravity would generate in a body falling through a height
equal to half the constant, a, of § 620: and this constant, a^ will
then be the length of the isochronous simple pendulum. Thus if
an elastic curve be held with its Hne of force vertical, and if a point,
jP, be moved along it with a constant velocity equal to Jga^ {a de-
noting the mean proportional between the radius of curvature at any
point, and its distance from the line of force,) the tangent at P will
keep always parallel to a simple pendulum, of length a^ placed at
any instant parallel to it, and projected with the same angular
velocity. Diagrams i to 5, correspond to vibrations of the pendu-
lum. Diagram 6 corresponds to the case in which the pendulum
would just reach its position of unstable equilibrium in an infinite
time. Diagram 7 corresponds to cases in which the pendulum flies
round continuously in one direction, with periodically increasing and
diminishing velocity. The extreme case, of the circular elastic
curve, corresponds to an infinitely long pendulum flying round with
finite angular velocity, which of course experiences only infinitely
small variation in the course of the revolution. A conclusion worthy
of remark is, that the rectification of the elastic curve is the same
analytical problem as finding the time occupied by a pendulum in
describing any given angle.
622. For the simple and important case of a natually straight
wire, acted on by a distribution of force, but not of couple, through
its length, the condition fulfilled at a perfectly free end, acted on by
neither force nor couple, is that the curvature is zero at the end, and
its rate of variation from zero, per unit of length from the end, is,
at the end, zero. In other words, the curvatures at points infinitely
near the end are as the squares of their distances from the end in
general (or, as some higher power of these distances, in singular
cases). The same statements hold for the change of curvature pro-
duced by the stress, if the unstrained wire is not straight, but the
other circumstances the same as those just specified.
623. As a very simple example of the equilibrium of a wire sub-
ject to forces through its length, let us suppose the natural form to
^ or THE
UNIVERSITY
Of .
STATICS OF SOLIDS AND FLUIDS.
333
234 ABSTRACT DYNAMICS.
be straight, and the appHed forces to be in lines, and the couples
to have their axes, all perpendicular to its length, and to be not great
enough to produce more than an infinitely small deviation from the
straight line. Further, in order that these forces and couples may-
produce no torsion, let the three flexure-torsion axes be perpendicular
to and along the wire. But we shall not hmit the problem further
by supposing the section of the wire to be uniform, as we should thus
exclude some of the most important practical applications, as to
beams of balances, levers in machinery, beams in architecture and
engineering. It is more instructive to investigate the equations of
equilibrium directly for this case than to deduce them from the
equations worked out above for the much more comprehensive
general problem. The particular principle for the present case is
simply that the rate of variation of the rate of variation, per unit of
length along the wire, of the bending couple in any plane through
the length, is equal, at any point, to the applied force per unit of
length, with the simple rate of variation of the applied couple sub-
tracted. This, together with the direct equations (§ 607) between
the component bending couples, gives the required equations of
equilibrium.
624. If the directions of maximum and minimum flexural rigidity
lie throughout the wire in two planes, the equations of equilibrium
become simplified when these planes are chosen as planes of re-
ference, XOY^ XOZ. The flexure in either plane then depends
simply on the forces in it, and thus the problem divides itself into
two quite independent problems of integrating the equations of
flexure in the two principal planes, and so finding the projections
of the curve on two fixed planes agreeing with their position when
the rod is straight.
625. When a uniform bar, beam, or plank is balanced on a single
trestle at its middle, the droop of its ends is only f of the droop
which its middle has when the bar is supported on trestles at its
ends. From this it follows that the former is f and latter f of the
droop or elevation produced by a force equal to half the weight of
the bar, applied vertically downwards or upwards to one end of it,
if the middle is held fast in a horizontal position. For let us first
suppose the whole to rest on a trestle under its middle, and let two
trestles be placed under its ends and gradually raised till the pressure
is entirely taken oft" from the middle. During this operation the
middle remains fixed and horizontal, while a force increasing to half
the weight, applied vertically upwards on each end, raises it through
a height equal to the sum of the droops in the two cases above
referred to. This result is of course proved directly by comparing
the absolute values of the droop in those two cases as found above,
with the deflection from the tangent at the end of the cord in the
elastic curve. Fig. 2, § 623, which is cut by the cord at right angles.
It may be stated otherwise thus : the droop of the middle of a
uniform beam resting on trestles at its ends is mcreased in the
STATICS OF SOLIDS AND FLUIDS. 235
ratio of 5 to 13 by laying a mass equal in weight to itself on its
middle : and, if the beam is hung by its middle, the droop of the
ends is increased in the ratio of 3 to 11 by hanging on each of
them a mass equal to half the weight of the beam.
626. The important practical problem of finding the distribution
of the weight of a solid on points supporting it, when more than
two of these are in one vertical plane, or when there are more than
three altogether, which (§ 588) is indeterminate^ if the soHd is
perfectly rigid, may be completely solved for a uniform elastic beam,
naturally straight, resting on three or more points in rigorously fixed
positions all nearly in one horizontal line, by means of the preceding
results.
If there are i points of support, the /— i parts of the rod between
them in order and the two end parts will form / + i curves expressed
by distinct algebraic equations, each involving four arbitrary con-
stants. For determining these constants we have 4/+ 4 equations in
all, expressing the following conditions : —
I. The ordinates of the inner ends of the projecting parts of the
rod, and of the two ends of each intermediate part, are respectively
equal to the given ordinates of the corresponding points of support
[2/ equations].
II. The curves on the two sides of each support have coincident
tangents and equal curvatures at the point of transition from one
to the other [2/ equations].
III. The curvature and its rate of variation per unit of length
along the rod, vanish at each end [4 equations].
Thus the equation of each part of the curve is completely de-
termined: and by means of it we find the shearing force in any
normal section. The difference between these in the neighbouring
portions of the rod on the two sides of a point of support, is of
course equal to the pressure on this point.
627. The solution for the case of this problem in which two
of the points of support are at the ends, and the third midway
between them either exactly in the line joining them, or at any
given very small distance above or below it, is found at once, without
analytical work, from the particular results stated in § 625. Thus
if we suppose the beam, after being first supported wholly by trestles
at its ends, to be gradually pressed up by a trestle under its
middle, it will bear a force simply proportional to the space through
which it is raised from the zero point, until all the weight is taken
off the ends, and borne by the middle. The whole distance through
which the middle rises during this process is, as we found, ^ . -^ ;
and this whole elevation is f of the droop of the middle in the
1 It need scarcely be remarked that indeterminateness does not exist in nature.
How it may occur in the problems of abstract dynamics, and is obviated by taking
something more of the properties of matter into account, is instructively illustrated
by the circumstances referred to in the text.
236 ABSTRACT DYNAMICS.
first position. If therefore, for instance, the middle trestle be fixed
exactly in the line joining those under the ends, it will bear f of
the whole weight, and leave y\ to be borne by each end. And
if the middle trestle be lowered from the line joining the end
ones by -/^ of the space through which it would have to be lowered
to relieve itself of all pressure, it will bear just \ of the whole weight,
and leave the other two-thirds to be equally borne by the two ends.
628. A wire of equal flexibility in all directions, and straight
when freed from stress, offers, when bent and twisted in any manner
whatever, not the slightest resistance to being turned round its elastic
central curve, as its conditions of equilibrium are in no way affected
by turning the whole wire thus equally throughout its length. The
useful application of this principle, to the maintenance of equal
angular motion in two bodies rotating round different axes, is
rendered somewhat difficult in practice by the necessity of a perfect
attachment and adjustment of each end of the wire, so as to have
the tangent to its elastic central curve exactly in line with the
axis of rotation. But if this condition is rigorously fulfilled, and
the wire is of exactly equal flexibility in every direction, and exactly
straight when free from stress, it will give, against any constant
resistance, an accurately uniform motion from one to another of
two bodies rotating round axes which may be inclined to one
another at any angle, and need not be in one plane. If they are
in one plane, if there is no resistance to the rotatory motion, and
if the action of gravity on the wire is insensible, it will take some
of the varieties of form (§ 620) of the plane elastic curve of James
Bernoulli. But however much it is altered from this, whether by
the axes not being in one plane, or by the torsion accompanying
the transmission of a couple from one shaft to the other, and
necessarily, when the axes are in one plane, twisting the wire out
of it, or by gravity, the elastic central curve will remain at rest,
the wire in every normal section rotating round it with uniform
angular velocity, equal to that of each of the two bodies which it
connects. Under Properties of Matter, we shall see, as indeed
may be judged at once from the performances of the vibrating
spring of a chronometer for twenty years, that imperfection in the
elasticity of a metal wire does not exist to any such degree as to
prevent the practical application of this principle, even in mechanism
required to be durable.
It is right to remark, however, that if the rotation be too rapid, the
equilibrium of the wire rotating round its unchanged elastic central
curve may become unstable, as is immediately discovered by experi-
ments (leading to very curious phenomena), when, as is often done in
illustrating the kinetics of ordinary rotation, a rigid body is hung by
a steel wire, the upper end of which is kept turning rapidly.
629. The definitions and investigations regarding strain, §§ 135-
161, constitute a kinematical introduction to the theory of elastic
solids. We must now, in commencing the elementary dynamics
STATICS OF SOLIDS AND FLUIDS
237
of the subject, consider the forces called into play through the
interior of a solid when brought into a condition of strain. We
adopt, from Rankine', the term stress to designate such forces,
as distinguished from strain defined (§ 135) to express the merely
geometrical idea of a change of volume or figure.
630. When through any space in a body under the action of force,
the mutual force between the portions of matter on the two sides of
any plane area is equal and parallel to the mutual force across any
equal, similar, and parallel plane area, the stress is said to be homo-
geneous through that space. In other words, the stress experienced
by the matter is homogeneous through any space if all equal, similar,
and similarly turned portions of matter within this space are similarly
and equally influenced by force.
631. To be able to find the distribution of force over the surface
of any portion of matter homogeneously stressed, we must know the
direction, and the amount per unit area, of the force across a plane
area cutting through it in any direction. Now if we know this for
any three planes, in three different directions, we can find it for a
plane in any direction as we see in a moment by considering what
is necessary for the equilibrium of a tetrahedron of the substance. The
resultant force on one of its sides must be equal and opposite to the
resultant of the forces on the three others, which is known if these sides
are parallel to the three planes for each of which the force is given.
632. Hence the stress, in a body homogeneously stressed, is com-
pletely specified when the direction, and the amount per unit area,
of the force on each of three distinct planes is given. It is, in the
analytical treatment of the subject, generally convenient to take these
planes of reference at right angles to one another. But we should
immediately fall into error did we not remark that the specification
here indicated consists not of nine but in reality only of six, inde-
pendent elements. For if the equilibrating forces on the six faces of
a cube be each resolved into three components parallel to its three
edges, OX, OV, OZ, we have in
all 18 forces; of which each pair
acting perpendicularly on a pair of
opposite faces, being equal and ^^
directly opposed, balance one an- ^,^^1 - ^k1 — ^->
other. The twelve tangential com-
ponents that remain constitute three
pairs of couples having their axes . ,
in the direction of the three edges, I i ..-Y
each of which must separately be in T
equilibrium. The diagram shows Q
the pair of equilibrating couples
having OV for axis; from the con-
sideration of which we infer that the
1 Canibrixigc and Dublin Mathematical Journal y 1850.
238 ABSTRACT DYNAMICS,
forces on the faces {zy)^ parallel to OZ, are equal to the forces on the
faces {yx)^ parallel to OX. Similarly, we see that the forces on the
faces {yx)^ parallel to OY, are equal to those on the faces (jcs), parallel
to OZ) and that the forces on {xz), parallel to OX^ are equal to
those on {zy), parallel to OY.
633. Thus, any three rectangular planes of reference being chosen,
we may take six elements thus, to specify a stress : T', Q, ^ the
normal components of the forces on these planes; and S, T, U
the tangential components, respectively perpendicular to OX, of the
forces on the two planes meeting in OX, perpendicular to OY, of
the forces on the planes meeting in 6^ 1^ and perpendicular to OZ,
of the forces on the planes meeting in OZ; each of the six forces
being reckoned, per unit of area. A normal component will be
reckoned as positive when it is a traction tending to separate
the portions of matter on the two sides of its plane. P, Q, R are
sometimes called simple longitudinal stresses, and »S, T, U simple
shearing stresses.
From these data, to find in the manner explained in § 631, the
force on any plane, specified by /, m, 71, the direction-cosines of
its normal ; let such a plane cut OX, OY, OZ in the three points
X, Y, Z. Then, if the area XYZ be denoted for a moment by A,
the areas YOZ, ZOX, XO Y, being its projections on the three rec-
tangular planes, will be respectively equal to Al, Am, An. Hence,
for the equilibrium of the tetrahedron of matter bounded by those
four triangles, we have, if F, G, H denote the components of the
force experienced by the first of them, XYZ, per unit of its area,
F.A =P.lA-v U. mA + T.nA,
and the two symmetrical equations for the components parallel to
6>Fand OZ. Hence, dividing by A, we conclude
F = Fl + Um-hTn]
G=Ul+Q?n + Sn\. (i)
Fr= Tl + Sm + Rn\
These expressions stand in the well-known relation to the ellipsoid
Px" + (2/ + i?^' + 2 {Syz + Tzx + Uxy) = i, (2)
according to which, if we take
X = lr, y = mr, z = nr,
and if X, /x, v denote the direction-cosines and p the length of the
perpendicular from the centre to the tangent plane at {x, y, z) of the
ellipsoidal surface, we have
^-Jr^ ^-Jr^ ^=Jr'
We conclude that
634. For any fully specified state of stress in a solid, a quadratic
surface may always be determined, which shall represent the stress
graphically in the following manner : —
To find the direction and the amount per unit area, of the force
STATICS OF SOLIDS AND FLUIDS. 239
acting across any plane in the solid, draw a line perpendicular to
this plane from the centre of the quadratic to its surface. The
required force will be equal to the reciprocal of the product of the
length of this line into the perpendicular from the centre to the
tangent plane at the point of intersection, and will be perpendicular
to the latter plane.
635. From this it follows that for any stress whatever there are
three determinate planes at right angles to one another such that the
force acting in the solid across each of them is precisely perpendicular
to it. These planes are called the principal or normal planes of the
stress; the forces upon them, per unit area, — its principal or normal
tractions; and the Hues perpendicular to them, — its principal or
normal axes, or simply its axes. The three principal semi-diameters
of the quadratic surface are equal to the reciprocals of the square
roots of the normal tractions. If, however, in any case each of the
three normal tractions is negative, it will be convenient to reckon
them rather as positive pressures; the reciprocals of the square roots
of which will be the semi-axes of a real stress-ellipsoid representing
the distribution of force in the manner explained above, with pressure
substituted throughout for traction.
636. When the three normal tractions are all of one sign, the
stress-quadratic is an ellipsoid; the cases of an ellipsoid of revolution
and a sphere being included, as those in which two, or all three, are
equal. When one of the three is negative and the two others posi-
tive, the surface is a hyperboloid of one sheet. When one of the
normal tractions is positive and the two others negative, the surface
is a hyperboloid of two sheets.
637. When one of the three principal tractions vanishes, while
the other two are finite, the stress-quadratic becomes a cylinder,
circular, elliptic, or hyperbolic, according as the other two are equal,
unequal of one sign, or of contrary signs. When two of the three
vanish, the quadratic becomes two planes; and the stress in this case
is (§ ^Z2)) called a simple longitudinal stress. The theory of prin-
cipal planes, and normal tractions just stated (§ 635), is then equiva-
lent to saying that any stress whatever may be regarded as made up
of three simple longitudinal stresses in three rectangular directions.
The geometrical interpretations are obvious in all these cases.
638. The composition of stresses is of course to be effected by
adding the component tractions thus: — If (F^, Q^, F^, S^, T^, U^),
(F^j (2^, F^, S^, 7;, U^), etc., denote, according to § 633, any given
set of stresses acting simultaneously in a substance, their joint effect
is the same as that of a simple resultant stress of which the specifica-
tion in corresponding terms is (IF, 2(2, '^F, 2^, 27", 26^.
639. Each of the statements that have now been made (§§ 630,
638) regarding stresses, is applicable to i7ifinitely small strains, if for
traction perpendicular to any plane, reckoned per unit of its area,
we substitute elongation, in the lines of the traction, reckoned per
unit of length; and for half the tangential traction parallel to any
240
ABSTRACT DYNAMICS.
direction, shear in the same direction, reckoned in the manner ex-
plained in § 154. The student will find it a useful exercise to study
in detail this transference of each one of those statements, and to
justify it by modifying in the proper manner the results of §§ 150, 151,
152, 153, i54» 161, to adapt them to infinitely small strains. It
must be remarked that the strain-quadratic thus formed according to
the rule of § 634, which may have any of the varieties of character
mentioned in §§ 6t^(), 637, is not the same as the strain-ellipsoid of
§ 141, which is always essentially an ellipsoid, and which, for an in-
finitely small strain, differs infinitely little from a sphere.
The comparison of § 151, with the result of § 632 regarding tan-
gential tractions is particularly interesting and important.
640. The following tabular synopsis of the meaning of the
elements constituting the corresponding rectangular specifications of
a strain and stress explained in preceding sections, will be found
convenient : —
co^r
Strain.
)onents
the
stress.
€
f
P
Q
a
s
b
T
Planes; of which
relative motion, or
across which force
is reckoned.
Direction
of relative
motion or
of force.
yz
zx
xy
X
y
z
(yx
\zx
y
z
(zy
[xy
z
X
(xz
X
\yz
y
u
641. If a unit cube of matter under any stress {P^ Q, P, S, T, U)
experience the infinitely small simple longitudinal strain e alone, the
work done on it will be Fe; since, of the component forces, F, U, T
parallel to OX^ U and T do no work in virtue of this strain. Simi-
larly, (2/, Fg are the works done if, the same stress acting, the simple
longitudinal strains f ox g are experienced, either alone. Again, if
the cube experiences a simple shear, a, whether we regard it (§ 151)
as a differential sliding of the planes yx, parallel to y, or of the planes
zx, parallel to z, we see that the work done is Sa: and similarly,
Tb if the strain is simply a shear b, parallel to OZ, of planes zy, or
parallel to OX, of planes xy. and Uc if the strain is a shear c, parallel
to OX, of planes xz, or parallel to OY, of planes yz. Hence the
whole work done by the stress {F, Q, F, S, T, U') on a, unit cube
taking the strain (e, f, g, a, b, c), is
Fe+ Q/+ Fg+Sa + Tb + Uc. (3)
It is to be remarked that, inasmuch as the action called a stress is
a system of forces which balance one another if the portion of
matter experiencing it is rigid, it cannot do any work when the
STATICS OF SOLIDS AND FLUIDS. 241
matter moves in any way without change of shape : and therefore no
amount of translation or rotation of the cube taking place along with
the strain can render the amount of work done different from that
just found.
If the side of the cube be of any length/, instead of unity, each
force will be /^ times, and each relative displacement/ times, and,
therefore, the work done p^ times the respective amounts reckoned
above. Hence a body of any shape, and of cubic content C, sub-
jected throughout to a uniform stress (P, Q, F, S, T, U) while taking
uniformly throughout a strain {e,f,g, a, b, c), experiences an amount
of work equal to
{Fe+ Q/+ Fg-hSa+n+ Uc)C. (4)
It is to be remarked that this is necessarily equal to the work done
on the bounding surface of the body by forces applied to it from
without. For the work done on any portion of matter within the
body is simply that done on its surface by the matter touching it all
round, as no force acts at a distance from without on the interior
substance. Hence if we imagine the whole body divided into any
number of parts, each of any shape, the sum of the work done on
all these parts is, by the disappearance of equal positive and negative
terms expressing the portions of the work done on each part by the
contiguous parts on all its sides, and spent by these other parts in
this action, reduced to the integral amount of work done by force
from without applied all round the outer surface.
642. If, now, we suppose the body to yield to a stress {P, Q, F^
S, T, U), and to oppose this stress only with its innate resistance to
change of shape, the differential equation of work done will [by (4)
with de, dfy etc., substituted for ^,/, etc.] be
dw = Fde + Qdf+ Fdg + Sda + Tdb + Udc. (5)
If w denote the whole amount of work done per unit of volume in
any part of the body while the substance in this part experiences a
strain (f,/, g, a, b, c) from some initial state regarded as a state of
no strain. This equation, as we shall see later, under Properties of
Matter, expresses the work done in a natural fluid, by distorting
stress (or difference of pressure indifferent directions) working against
its innate viscosity; and w is then, according to Joule's discovery,
the dynamic value of the heat generated in the process. The equa-
tion may also be applied to express the work done in straining an
imperfectly elastic sohd, or an elastic solid of which the temperature
varies during the process. In all such applications the stress will
depend partly on the speed of the straining motion, or on the varying
temperature, and not at all, or not solely, on the state of strain at any
moment, and the system will not be dynamically conservative.
643. Definition. — A perfectly elastic body is a body which, when
brought to any one state of strain, requires at all times the same
stress to hold it in this state; however long it be kept strained, or
however rapidly its state be altered from any other strain, or from
no strain, to the strain in question. Here, according to our plan
T. 16
242 ABSTRACT DYNAMICS.
(§§ 396) 4°^) ^or Abstract Dynamics, we ignore variation of tempera-
ture in the body. If, however, we add a condition of absolutely no
variation of temperature, or of recurrence to one specified temperature
after changes of strain, we have a definition of that property of perfect
elasticity towards which highly elastic bodies in nature approximate;
and which is rigorously fulfilled by all fluids, and may be so by some
real soHds, as homogeneous crystals. But inasmuch as the elastic
reaction of every kind of body against strain varies with varying
temperature, and (a thermodynamic consequence of this, as we shall
see later) any increase or diminution of strain in an elastic body is
necessarily* accompanied by a change of temperature; even a per-
fectly elastic body could not, in passing through different strains,
act as a rigorously conservative system, but on the contrary, must
give rise to dissipation of energy in consequence of the conduction
or radiation of heat induced by these changes of temperature.
But by making the changes of strain quickly enough to prevent
any sensible equilization of temperature by conduction or radiation
(as, for- instance, Stokes has shown, is done in sound of musical
notes travelling through air); or by making them slowly enough to
allow the temperature to be maintained sensibly constant' by proper
appliances; any highly elastic, or perfectly elastic body in nature may
be got to act very nearly as a conservative system.
644. In nature, therefore, the integral amount, w, of work defined
as above, is for a perfectly elastic body, independent (§ 246) of the
series of configurations, or states of strain, through which it may have
been brought from the first to the second of the specified conditions,
provided it has not been allowed to change sensibly in temperature
during the process.
When the whole amount of strain is infinitely small, and the stress-
components are therefore all altered in the same ratio as the strain-
components if these are altered all in any one ratio; w must be a
homogeneous quadratic function of the six variables e, f, g, a, b, «f,
which, if we denote by {e, e), {/,/).,. (e,/). . . constants depending
on the quality of the substance and on the directions chosen for the
axes of co-ordinates, we may write as follows : —
W = U{e, e)^ + {/J)f ■^{g,g) g' + {a,a)a' + (3,b) d'+ (Cc)^
+ 2 (e,/) e/+ 2 {e,g) eg +2 {e,a)ea+2{e,b)eb+2 (e^ c)ec
+ 2 {f,g)fg^ 2 {f,a)fa + 2 {/,b)/b + 2 (/ c)/c
+ 2(g,a)ga + 2 (g, b)gb + 2 (g, c)gc
+ 2 [a, b)ab + 2 (a^c) ac
+ 2 {b^c)bc\
The 21 co-efficients (^, ^), (/,/)... (<^, <r), in this expression con-
stitute the 21 'co-efficients of elasticity,' which Green first showed to
be proper and essential for a complete theory of the dynamics of an
elastic sohd subjected to infinitely small strains. The only condition
^ *0n the Thermoelastic and Therm omagnetic Properties of Matter' (W.
Thomson). Quarterly Journal of Mathematics, K'^xW i^ll, ^ Ibid,
STATICS OF SOLIDS AND FLUIDS. 243
that can be theoretically imposed upon these co-efficients is that they
must not permit w to become negative for any values, positive or
negative, of the strain-components e, /,. . . c. Under Properties of
Matter, we shall see that a false theory (Boscovich's), falsely worked
out by mathematicians, has led to relations among the co-efficients of
elasticity which experiment has proved to be false.
645. The average stress, due to elasticity of the solid, when
strained from its natural condition to that of strain (e, /, g, «, b, c) is
(as from the assumed applicability of the principle of superposition
we see it must be) just half the stress required to keep it in this
state of strain.
646. A body is called homogeneous when any two equal, similar
parts of it, with corresponding lines parallel and turned towards
the same parts, are undistinguishable from one another by any
difference in quality. The perfect fulfilment of this condition with-
out any limit as to the smallness of the parts, though conceivable,
is not generally regarded as probable for any of the real solids or
fluids known to us, however seemingly homogeneous. It is, we
believe, held by all naturalists that there is a molecular structure^
according to which, in compound bodies such as water, ice, rock-
crystal, etc., the constituent substances lie side by side, or arranged
in groups of finite dimensions, and even in bodies called si77iple
(i.e. not known to be chemically resolvable into other substances)
there is no ultimate homogeneousness. In other words, the prevail-
ing belief is that every kind of matter with which we are acquainted
has a more or less coarse-grained texture, whether having visible
molecules, as great masses of solid stone or brick-building, or natural
granite or sandstone rocks; or, molecules too small to be visible
or directly measurable by us (but not infinitely small) ^ in seemingly
homogeneous metals, or continuous crystals, or liquids, or gases.
We must of course return to this subject under Properties of Matter ;
and in the meantime need only say that the definition of homogeneous-
ness may be applied practically on a very large scale to masses of
building or coarse-grained conglomerate rock, or on a more moderate
scale to blocks of common sandstone, or on a very small scale to
seemingly homogeneous metals^; or on a scale of extreme, undis-
covered fineness, to vitreous bodies, continuous crystals, sohdified
gums, as India rubber, gum-arabic, etc., and fluids.
647. The substance of a homogeneous solid i^ called isotropic
when a spherical portion of it, tested by any physical agency, exhibits
no difference in quality however it is turned. Or, which amounts
to the same, a cubical portion cut from any position in an isotropic
body exhibits the same qualities relatively to each pair of parallel
faces. Or two equal and similar portions cut from any positions
* Probably not undiscoverably smz\\, although of dimensions not yet known to us.
' Which, however, we know, as recently proved by Deville and Van Troost, are
porous enough at high temperature to allow very free percolation of gases.
16 — 2
244 ABSTRACT DYNAMICS.
in the body, not subject to the condition of parallelism (§ 646),
are undistinguishable from one another. A substance which is not
isotropic, but exhibits differences of quality in different directions,
is called aeolotropic.
648. An individual body, or the substance of a homogeneous
solid, may be isotropic in one quality or class of qualities, but
aeolotropic in others.
Thus in abstract dynamics a rigid body, or a group of bodies
rigidly connected, contained within and rigidly attached to a rigid
spherical surface, is kinetically symmetrical {§ 239) if its centre of
inertia is at the centre of the sphere, and if its moments of inertia
are equal round all diameters. It is also isotropic relatively to gravi-
tation if it is centrobaric (§ 542), so that the centre of figure is not
merely a centre of inertia, but a true centre of gravity. Or a trans-
parent substance may transmit light at different velocities in different
directions through it (that is, be doubly refracting), and yet a cube
of it may (and generally does in natural crystals) absorb the same
part of a beam of white hght transmitted across it perpendicularly
to any of its three pairs of faces. Or (as a crystal which exhibits
dichroisin) it may be aeolotropic relatively to the latter, or to either,
optic quality, and yet it may conduct heat equally in all directions.
649. The remarks of § 646 relative to homogeneousness in the
aggregate, and the supposed ultimately heterogeneous texture of all
substances however seemingly homogeneous, indicate corresponding
limitations and non-rigorous practical interpretations of isotropy.
650. To be elastically isotropic, we see first that a spherical or
cubical portion of any solid, if subjected to uniform normal pressure
(positive or negative) all round, must, in yielding, experience no
deformation : and therefore must be equally compressed (or dilated)
in all directions. But, further, a cube cut from any position in it,
and acted on by tange7itial or distorting stress (§ 633) in planes
parallel to two pairs of its sides, must experience simple deformation,
or shear (§ 150), in the same direction, unaccompanied by condensa-
tion or dilatation', and the same in amount for all the three ways
in which a stress may be thus applied to any one cube, and for
different cubes taken from any different positions in the solid.
651. Hence the elastic quality of a perfectly elastic, homogeneous,
isotropic solid is fully defined by two elements; — its resistance to
compression, and its resistance to distortion. The amount of uni-
form pressure in all directions, per unit area of its surface, required
to produce a stated very small compression, measures the first of
^ It must be remembered that the changes of figure and volume we are con-
cerned with are so small that the principle of superposition is applicable; so that
if any distorting stress produced a condensation, an opposite distorting stress
would produce a dilatation, which is a violation of the isotropic condition. But it
is possible that a distorting stress may produce, in a truly isotropic solid, conden-
sation or dilatation in proportion to the square of its value : and it is probable that
such effects may be sensible in India x-ubber, or cork, or other bodies susceptible
of great deformations or compressions, with persistent elasticity.
STATICS OF SOLIDS AND FLUIDS.
245
these, and the amount of the distorting stress required to produce
a stated amount of distortion measures the second. The numerical
measure of the first is the compressing pressure divided by the
diminution of the bulk of a portion of the substance which, when
uncompressed, occupies the unit volume. It is sometimes called
the elasticity of volume^ or the resistance to compression. Its reciprocal,
or the amount of compression on unit of volume divided by the
compressing pressure, or, as we may conveniently say, the compression
per unit of volume, per unit of compressing pressure, is commonly
called the compressibility. The second, or resistance to change of shape,
is measured by the tangential stress (reckoned as in § 62,2,) divided by
the amount of the distortion or shear (§ 154) which it produces, and
is called the rigidity of the substance, or its elasticity of figure.
652. From § 148 it follows that a strain compounded of a simple
extension in one set of parallels, and a simple contraction of equal
amount in any other set perpendicular to those, is the same as a
simple shear in either of the two sets of planes cutting the two
sets of parallels at 45°. And the numerical measure (§ 154) of this
shear, or simple distortion, is equal to dojible the amount of the
elongation or contraction (each measured, of course, per unit of
length). Similarly, we see (§ 639) that a longitudinal traction (or
negative pressure) parallel to one line, and an equal longitudinal
positive pressure parallel to any line at right angles to it, is equivalent
to a distorting stress of tangential tractions (§ 632) parallel to the
planes which cut those lines at 45°. And the numerical measure
of this distorting stress, being (§ 633) the amount' of the tangential
traction in either set of planes, is equal to the amount of the positive
or negative normal pressure, 7iot doubled.
653. Since then any stress whatever may be made up of simple
longitudinal stresses, it follows that, to find the relation between any
stress and the strain produced by
it, we have only to find the strain
produced by a single longitudinal
stress, which we may do at
once thus : — A simple longitudinal
stress, jP, is equivalent to a uni-
form dilating tension ^F in all
directions, compounded with two
distorting stresses, each equal to
^F, and having a common axis
in the line' of the given longitu-
dinal stress, and their other two
axes any two lines at right angles
to one another and to it. The
diagram, drawn in a plane through
one of these latter lines, and the
former, sufficiently indicates the synthesis; the only forces not shown
being those perpendicular to its plane.
246 ABSTRACT DYNAMICS.
Hence if n denote the rigidity, and k the reststa?ice to dilatation
[being the same as the reciprocal of the compressibility (§ 651)], the
effect will be an equal dilatation in all directions, amounting, per
unit of volume, to
1 P
compounded with two equal distortions, each amounting to
1 P
and having (§ 650) their axes in the directions just stated as those
of the distorting stresses.
654. The dilatation and two shears thus determined may be
conveniently reduced to simple longitudinal strains by still following
the indications of § 652, thus: —
The two shears together constitute an elongation amounting to
- — in the direction of the given force, jP, and equal contraction
amounting to 2— in all directions perpendicular to it. And the
\ p ^
cubic dilatation 2_ implies a lineal dilatation, equal in all directions,
\ p
amounting to ~- . On the whole, therefore, we have
linear elongation = P T — + — j , in the direction of the applied
stress, and
linear contraction =/* (-7 a ) » ^^ ^ directions perpendicular
to the applied stress.
(3)
655. Hence when the ends of a column, bar, or wire, of isotropic
material, are acted on by equal and opposite forces, it experiences
a lateral lineal contraction, equal to ---t r o^ the longitudinal
dilatation, each reckoned as usual per unit of lineal measure. One
specimen of the fallacious mathematics above referred to (§ 644), is
a celebrated conclusion of Navier's and Poisson's that this ratio
is J, which requires the rigidity to be f of the resistance to com-
pression, for all solids : and which was first shown to be false by
Stokes' from many obvious observations, proving enormous discre-
pancies from it in many well-known bodies, and rendering it most
improbable that there is any approach to a constancy of ratio between
^ ' On the Friction of Fluids in Motion, and the Equilibrium and Motion of
Elastic Solids.' Trans. Camb. Phil. Soc, April 1845. See also Camb. and
Dub. Math. Jour., March 1848.
STATICS OF SOLIDS AND FLUIDS 247
rigidity and resistance to compression in any class of solids. Thus
clear elastic jellies, and India rubber, present familiar specimens of
isotropic homogeneous solids, which, while differing very much from
one another in rigidity ('stiffness'), are probably all of v6ry nearly
the same compressibiHty as water. This being -g-^gVcru P^'^ pound
per square inch; the resistance to compression, measured by its
reciprocal, or, as we may read it, '308000 lbs. per square inch,*
is obviously many hundred times the absolute amount of the
rigidity of the stiffest of those substances. A column of any of
them, therefore, when pressed together or pulled out, within its limits
of elasticity, by balancing forces applied to its ends (or an India
rubber band when pulled out), experiences no sensible change of
volume, though a very sensible change of length. Hence the pro-
portionate extension or contraction of any transverse diameter must
be sensibly equal to J the longitudinal contraction or extension : and
for all ordinary stresses, such substances may be practically regarded
as incompressible elastic solids. Stokes gave reasons for believing
that metals also have in general greater resistance to compression, in
proportion to their rigidities, than according to the fallacious theory,
although for them the discrepancy is very much less than for the
gelatinous bodies. This probable conclusion was soon experimentally
demonstrated by Wertheim, who found the ratio of lateral to longi-
tudinal change of lineal dimensions, in columns acted on solely by
longitudinal force, to be about ^ for glass or brass; and by Kirchhoff,
who, by a very well-devised experimental method, found '387 as the
value of that ratio for brass, and '294 for iron. For copper we find
that it probably lies between "226 and '441, by recent experiments^
of our own, measuring the torsional and longitudinal rigidities (§§ 609,
657) of a copper wire.
656. All these results indicate rigidity less in proportion to the
compressibility than according to Navier's and Poisson's theory.
And it has been supposed by many naturalists, who have seen the
necessity of abandoning that theory as inapplicable to ordinary solids,
that it may be regarded as the proper theory for an ideal perfect solid,
and as indicating an amount of rigidity not quite reached in any
real substance, but approached to in some of the most rigid of
natural solids (as, for instance, iron). But it is scarcely possible
to hold a piece of cork in the hand without perceiving the fallacious-
ness of this last attempt to maintain a theory which never had any
good foundation. By careful measurements on columns of cork
of various forms (among them, cylindrical pieces cut in the ordinary
way for bottles) before and after compressing them longitudinally in a
Brahmah's press, we have found that the change of lateral dimensions
is insensible both with small longitudinal contractions and return
dilatations, within the limits of elasticity, and with such enormous
longitudinal contractions as to ^ or J of the original length. It is
thus proved decisively that cork is much more rigid, while metals,
1 'On the Elasticity and Viscosity of Metals' (W. Thomson), Froc, R.S., May
1865.
248 ABSTRACT DYNAMICS.
glass, and gelatinous bodies are all less rigid, in proportion to
resistance to compression, than the supposed 'perfect solid'; and the
utter worthlessness of the theory is experimentally demonstrated.
657. The modulus of elasticity of a bar, wire, fibre, thin filament,
band, or cord of any material (of which the substance need not be
isotropic, nor even homogeneous within one normal section), as a
bar of glass or wood, a metal wire, a natural fibre, an India rubber
band, or a common thread, cord, or tape, is a term introduced
by Dr. Thomas Young to designate what we also sometimes call its
longitudinal rigidity: that is, the quotient obtained by dividing the
simple longitudinal force required to produce any infinitesimal
elongation or contraction by the amount of this elongation or con-
traction reckoned as always per unit of length.
658. Instead of reckoning the modulus in units of weight, it is
sometimes convenient to express it in terms of the weight of the unit
length of the rod, wire, or thread. The modulus thus reckoned,
or, as it is called by some writers, the length of the modulus, is
of course found by dividing the weight-modulus by the weight of
the unit length. It is useful in many applications of the theory of
elasticity; as, for instance, in this result, which will be proved
later: — the velocity of transmission of longitudinal vibrations (as of
sound) along a bar or cord, is equal to the velocity acquired by
a body in falling from a height equal to half the length of the
modulus*.
659. The specific modulus of elasticity of an isotropic substance^ or,
as it is most often called, simply the modulus of elasticity of the sub-
sta?ice, is the modulus of elasticity of a bar of it having some definitely
specified sectional area. If this be such that the weight of unit
length is unity, the modulus of the substance will be the same as the
length of the modulus of any bar of it; a system of reckoning which,
as we have seen, has some advantages in application. It is, how-
ever, more usual to choose a common unit of area as the sectional
area of the bar referred to in the definition. There must also be a
definite understanding as to the unit in terms of which the force is
measured, which may be either the absolute u?iit (§ i88): or the
gravitation unit for a specified locality; that is (§ 191), the weight in
that locality of the unit of mass. Experimenters hitherto have stated
their results in terms of the gravitation unit, each for his own locality;
the accuracy hitherto attained being scarcely in any cases sufficient to
^ It is to be understood that the vibrations in question are so much spread out
through the length of the body, that inertia does not sensibly influence the trans-
verse contractions and dilatations which (unless the substance have in this respect
the peculiar character presented by cork, § 656) take place along with them. Also,
under Thermodynamics, we shall see that changes of temperature produced by the
varying strains cause changes of stress which, in ordinary solids, render the velocity
of transmission of longitudinal vibrations sensibly greater than that calculated by
the rule stated in the text, if we use the static viodtdus as understood from the
definition there given; and we shall learn to take into account the thermal effect
])yusing a definite static modulus, ox kinetic modulus, according to the circumstances
of any case that may occur. '
STATICS OF SOLIDS AND FLUIDS, 249
require corrections for the different forces of gravity in the different
places of observation.
660. The most useful and generally convenient specification of
the modulus of elasticity of a substance is in grammes-weight per
square centimetre. This has only to be divided by the specific
gravity of the substance to give the length of the modulus. British
measures, however, being still unhappily sometimes used in practical
and even in high scientific statements, we may have occasion to refer
to reckonings of the modulus in pounds per square inch or per square
foot, or to length of the modulus in feet.
661. The reckoning most commonly adopted in British treatises
on mechanics and practical statements is pounds per square inch.
The modulus thus stated must be divided by the weight of 1 2 cubic
inches of the soHd, or by the product of its specific gravity into '4337 \
to find the length of the modulus, in feet.
To reduce from pounds per square inch to grammes per square
centimetre, multiply by 70-3 1, or divide by -014223. French engineers
generally state their results in kilogrammes per square millimetre,
and so bring them to more convenient numbers, being x^j-oVuo- ^^ ^^
inconveniently large numbers expressing moduli in grammes-weight
per square centimetre.
662. The same statements as to units, reducing factors, and nominal
designations, are applicable to the resistance to compression of any
elastic solid or fluid, and to the rigidity (§ 651) of an isotropic body;
or, in general, to any one of the 2 1 co-efficients in the expressions
for terms in stresses of strains, or to the reciprocal of any one of
the 21 co-efficients in the expressions for strains in terms of stresses,
as well as to the modulus defined by Young.
663. In §§ 652, 653 we examined the effect of a simple longitudinal
stress, in producing elongation in its own direction, and contraction
^ This decimal being the weight in pounds of 12 cubic inches of water. The
one great advantage of the French metrical system is, that the mass of the unit
volume (i cubic centimetre) of water at its temperature of maximum density
(3° '945 C-) is unity (i gramme) to a sufficient degree of approximation for almost
all practical purposes. Thus, according to this system, tlie density of a body and
its specific gravity mean one and the same thing ; whereas on the British no-system
the density is expressed by a number found by multiplying the specific gravity by
one number or another, according to the choice of a cubic inch, cubic foot, cubic
yard, or cubic mile that is made for the unit of volume; and the grain, scruple,
gunmaker's drachm, apothecary's drachm, ounce Troy, ounce avoirdupois, pound
Troy, pound avoirdupois, stone (Imperial, Ayrshire, Lanarkshire, Dumbarton-
shire), stone for hay, stone for corn, quarter (of a hundredweight), quarter (of
corn), hundredweight, or ton, that is chosen for unit of mass. It is a remarkable
phenomenon, belonging rather to moral and social than to physical science, that
a people tending naturally to be regulated by common sense should voluntarily
condemn themselves, as the British have so long done, to unnecessary hard labour
in every action of common business or scientific work related to measurement, from
which all the other nations of Europe have emancipated themselves. We have
been informed, through the kindness of Professor W. H. Miller, of Cambridge,
that he concludes, from a very tmstworthy comparison of standards by Kupflfer, of
St. Petersburgh, that the weight of a cubic decimetre of water at temperature of -
maximum density is 1000*013 grammes.
250
ABSTRACT DYNAMICS.
in lines perpendicular to it. With stresses substituted for strains, and
strains for stresses, we may apply the same process to investigate the
longitudinal and lateral tractions required to produce a simple longi-
tudinal strain (that is, an elongation in one direction, with no change
of dimensions perpendicular to it) in a rod or solid of any shape.
Thus a simple longitudinal strain e is equivalent to a cubic dilata-
tion e without change of figure (or linear dilatation \e equal in all
directions), and two distortions consisting each of dilatation \e in the
given direction, and contraction \e in each of two directions perpen-
dicular to it and to one another. To produce the cubic dilatation, <?,
alone requires (§ 651) a normal traction ke equal in all directions.
And, to produce either of the distortions simply, since the measure
(§ 154) of each is \e^ requires a distorting stress equal to « x §<?, which
consists of tangential tractions each equal to this amount, positive (or
drawing outwards) in the line of the given elongation, and negative (or
pressing inwards) in the perpendicular direction. Thus we have in all
normal traction = (>^ + 1^«) ^, in the direction of the given*]
strain, and I
normal traction = (^-f/2) <f, in every direction perpen-|
dicular to the given strain. J
(4)
664. If now we suppose any possible infinitely small strain (^,/, g,
a, d, c), according to the specification of § 640, to be given to a body,
the stress {!*, Qj R, S, T, U) required to maintain it will be expressed
by the following formulae, obtained by successive applications of
§ ddTy (4) to the components e^ f, g separately, and of § 651 to
a, b, c—
S= na, T== nb, U= nc^
F=^%e + ^{f+g\
Q = ni/+^(g + e),
R = Ug+^(e+f),
where
(5)
665. Similarly, by § 651 and § 654 (3), we have
a=^-S,b=-T,c = - C/,'
Mg={R-^{F+Q)},
gnk
where
and
M=
(6)
STATICS OF SOLIDS AND FLUIDS. 251
as the formulae expressing the strain (^, /, g, a, b, c) in terms of
the stress (F, (2, F, S, T, U). They are of course merely the
algebraic inversions of (5) ; and they might have been found by
solving these for e, / g^ a, b, c, regarded as the unknown quantities.
M is here introduced to denote Young's modulus.
666. To express the equation of energy for an isotropic substance,
we may take the general formula,
'W = l{Fe-¥ Q/+Fg+Sa + Tb + L/c),
and eliminate from it F, Q, etc., by (5) of § 664, or, again, e, /, etc.,
by (6) of § 665, we thus find
667. The mathematical theory of the equilibrium of an elastic
solid presents the following general problems :
A solid of any given shape, when undisturbed, is acted on in its
substance by force distributed through it in any given manner, and dis-
placements are arbitrarily produced, or forces arbitrarily applied, over
its bounding surface. It is required to find the displacement of every
poi?tt of its substance.
This problem has been thoroughly solved for a shell of homo-
geneous isotropic substance bounded by surfaces which, when undis-
turbed, are spherical and concentric ; but not hitherto for a body
of any other shape. The limitations under which solutions have
been obtained for other cases (thin plates and rods), leading, as we
have seen, to important practical results, have been stated above
(§ 605). To demonstrate the laws (§ 607) which were taken in
anticipation will also be one of our applications of the general
equations for interior equilibrium of an elastic solid, which we now
proceed to investigate.
668. Any portion in the interior of an elastic solid may be
regarded as becoming perfectly rigid (§ 584) without disturbing the
equilibrium either of itself or of the matter round it. Hence the traction
exerted by the matter all round it, regarded as a distribution of force
applied to its surface, must, with the applied forces acting on the sub-
stance of the portion considered, fulfil the conditions of equilibrium of
forces acting on a rigid body. This statement, applied to an infinitely
small rectangular parallelepiped of the body, gives the general differ-
ential equations of internal equilibrium of an elastic solid. It is to be
remarked that three equations suffice ; the conditions of equilibrium
for the couples being secured by the relation estabHshed above (§632)
among the six pairs of tangential component tractions on the six
faces of the figure.
669. One of the most beautiful applications of the general equa-
tions of internal equilibrium of an elastic solid hitherto made is
252 ABSTRACT DYNAMICS.
that of M. de St. Venant to 'the torsion of prisms ^' To one
end of a long straight prismatic rod, wire, or solid or hollow cylinder
of any form, a given couple is applied in a plane perpendicular to
the length, while the other end is held fast : it is required to find
the degree of twist produced, and the distribution of strain and
stress throughout the prism. The conditions to be satisfied here
are that the resultant action between the substance on the two sides
of any normal section is a couple in the normal plane, equal to the
given couple. Our work for solving the problem will be much
simplified by first establishing the following preliminary propo-
sitions : —
670. Let a solid (whether aeolotropic or isotropic) be so acted
on by force applied from without to its boundary, that throughout its
interior there is no normal traction on any plane parallel or per-
pendicular to a given plane, XOY^ which implies, of course, that
there is no distorting stress with axes in or parallel to this plane, and
that the whole stress at any point of the solid is a simple distorting
stress of tangential forces in some direction in the plane parallel to
XO Vy and in the plane perpendicular to this direction. Then —
(i) The interior distorting stress must be equal, and similarly
directed, in all parts of the solid lying in any line perpendicular
to the plane XO Y.
(2) It being premised that the traction at every point of any
surface perpendicular to the plane XOY'is, by hypothesis, a distribu-
tion of force in lines perpendicular to this plane ; the integral amount
of it on any closed prismatic or cylindrical surface perpendicular to
XO Y, and bounded by planes parallel to it, is zero.
(3) The matter within the prismatic surface and terminal planes of
(2) being supposed for a moment (§ 584) to be rigid, the distribution
of tractions referred to in
(2) constitutes a couple
whose moment, divided by
the distance between those
terminal planes, is equal to
the resultant force of the
tractions on the area of
either, and whose plane is
parallel to the lines of these
resultant forces. In other
^ words, the moment of the
O ~~ X" distribution of forces over
the prismatic surface referred to in (2) round any line (OY or OX) in
the plane XOY, is equal to the sum of the components {T or *S),
perpendicular to the same line, of the traction in either of the
terminal planes multiplied by the distance between these planes.
^ Memoires des Savants ^Irangas, 1855. ' De 1^ Torsion des Prismes, avec des
Considerations sur leur Flexion,' etc.
STATICS OF SOLIDS AND FLUIDS 253
To prove (i) consider for a moment as rigid (§ 584) an infinite-
simal prism, AB (of sectional area to), per- ^
pendicular to XOY, and having plane ends, f"^ ^-lu)
A^ B, parallel to it. There being no forces
on its sides (or cylindrical boundary) per-
pendicular to its length, its equilibrium so far
as motion in the direction of any line {OX),
perpendicular to its length, requires that the
components of the tractions on its ends be
equal and in opposite directions. Hence,
in the notation § 633, the distorting - stress
components, T, must be equal at A and B -, Yu--^-
and so must the stress components S, for the B
same reason.
To prove (2) and (3) we have only to remark that they are
required for the equilibrium of the rigid prism referred to
in (3).
671. For a soUd or hollow circular cylinder, the solution of § 669
(given first, we believe, by Coulomb) obviously is that each circular
normal section remains unchanged in its own dimensions, figure,
and internal arrangement (so that every straight line of its particles
remains a straight line of unchanged length), but is turned round
the axis of the cylinder through such an angle as to give a uniform
rafe of twist equal to the applied couple divided by the product
of the moment of inertia of the circular area (whether annular or
complete to the centre) into the rigidity of the substance.
672. Similarly, we see that if a cylinder or prism of any shape
be compelled to take exactly the state of strain above specified (§671)
with the line through the centres of inertia of the normal sections,
taken instead of the axis of the cylinder, the mutual action between
the parts of it on the two sides of any normal section will be a couple
of which the moment will be expressed by the same formula, that is,
the product of the rigidity, into the rate of twist, into the moment
of inertia of the section round its centre of inertia.
673. But for any other shape of prism than a solid or symmetrical
hollow circular cylinder, the supposed state of strain will require,
besides the terminal opposed couples, force parallel to the length
of the prism, distributed over the prismatic boundary, in proportion
to the distance along the tangent, from each point of the surface,
to the point in which this Hne is cut by a perpendicular to it from the
centre of inertia of the normal section. To prove this let a normal
section of the prism be represented in the annexed diagram (page 254).
Let PK representing the shear at any point, P, close to the prismatic
boundary, be resolved into PN and FT respectively along the nor-
mal and tangent. The whole shear, PK, being equal to rr, its
component, /'iV^, is equal to rr sin w or t./^^. The corresponding
component of the required stress is nr.PE, and involves (§ 632)
equal forces in the plane of the diagram, and in the plane through
254 ABSTRACT DYNAMICS.
TP perpendicular to it, each amounting to nr.PE per unit of
area.
An application of force
equal and opposite to the
distribution thus found
over the prismatic boun-
dary, would of course alone
r produce in the prism, other-
wise free, a state of strain
which, compounded with
that supposed above,would
give the state of strain ac-
tually produced by the sole
application of balancing
couples to the two ends.
The result, it is easily seen (and it will be proved below), consists of
an increased twist, together with a warping of naturally plane normal
sections, by infinitesimal displacements perpendicular to themselves,
into certain surfaces of anticlastic curvature, with equal opposite
curvatures in the principal sections (§ 122) through every point.
This theory is due to St. Venant, who not only pointed out the falsity
of the supposition admitted by several previous writers, that Cou-
lomb's law holds for other forms of prism than the solid or hollow
circular cylinder, but discovered fully the nature of the requisite
correction, reduced the determination of it to a problem of pure
mathematics, worked out the solution for a great variety of important
and curious cases, compared the results with observation in a manner
satisfactory and interesting to the naturalist, and gave conclusions
of great value to the practical engineer.
674. We take advantage of the identity of mathematical conditions
in St. Venant's torsion problem, and a hydrokinetic problem first
solved a few years earlier by Stokes \ to give the following statement,
which will be found very useful in estimating deficiencies in torsional
rigidity below the amount calculated from the fallacious extension
of Coulomb's law : —
675. Conceive a liquid of density n completely filling a closed
infinitely light prismatic box of the same shape within as the given
elastic prism and of length unity, and let a couple be applied to the
box in a plane perpendicular to its length. The effective moment
of inertia of the liquid^ will be equal to the correction by which the
torsional rigidity of the elastic prism calculated by the false extension
of Coulomb's law, must be diminished to give the true torsional
rigidity.
Further, the actual shear of the solid, in any infinitely thin plate of
* 'On some cases of Fluid Motion,' Cambridge Philosophical Transactions , 1843.
' That is the moment of inertia of a rigid solid which, as will be proved in
Vol. II., may be fixed within the box, if the liquid be removed, to make its
motions the same as they are with the liquid in it.
STATICS OF SOLIDS AND FLUIDS.
255
it between two normal sections, will at each point be, when reckoned
as a differential sliding (§ 151) parallel to their planes, equal to and
in the same direction as the velocity of the liquid relatively to the
containing box.
676. St. Venant's treatise abounds in beautiful and instructive
graphical illustrations of his results, from which we select the
following : —
(i) Elliptic cylinder. The plain and dotted curvilineal arcs are
* contour lines ' {coupes topographiques) of the section as warped by
torsion ; that is to say, lines in which it is cut by a series of parallel
planes, each perpendicular to the axis. These hues are equilateral
hyperbolas in this case. The arrows indicate the direction of rotation
in the part of the prism above the plane of the diagram.
(2) Equilateral triangular prism, — The contour lines are shown
as in case (i); the dotted curves being those where the warped
section falls below the plane of the diagram, the direction of rotation
256
ABSTRACT DYNAMICS.
of the part of the prism above the plane being indicated by the
bent arrow.
(3) This diagram shows a series of lines given by St. Venant,
and more or less resembling squares, their common equation
containing only one constant a. It is remarkable that the values
fl! = o-5 and a = -\{j2-\) give similar but not equal curvi-
lineal squares (hollow sides and acute angles), one of them turned
through half a right angle relatively to the other. Everything in the
diagram outside the larger of these squares is to be cut away as
irrelevant to the physical problem; the series of closed curves
remaining exhibits figures of prisms, for any one of which the torsion
problem is solved algebraically. These figures vary continuously from
a circle, inwards to one of the acute-angled squares, and outwards to
the other : each, except these extremes, being a continuous closed
curve with no angles. The curves for a = 0-4 and a = -0-2 approach
remarkably near to the rectilineal squares, partially indicated in the
diagram by dotted lines.
(4) This diagram shows the contour lines, in all respects as in the
cases (i) and (2) for the case of a prism having for section the figure
indicated. The portions of curve outside the continuous closed curve
are merely indications of mathematical extensions irrelevant to the
physical problem.
STATICS OF SOLIDS AND FLUIDS.
Y
257
(5) This shows, as in the other cases, the contour lines for the
warped section of a square prism under torsion.
r
T.
17
258
ABSTRACT DYNAMICS.
(6), (7), (8). These are shaded drawings, showing the appear-
ances presented by elliptic, square, and flat rectangular bars under
exaggerated torsion, as may be realized with such a substance as
India rubber.
677. Inasmuch as the moment of inertia of a plane area about
an axis through its centre of inertia perpendicular to its plane is
obviously equal to the sum of its moments of inertia round any two
axes through the same point, at right angles to one another in its
plane, the fallacious extension of Coulomb's law, referred to in § 673,
71
would make the torsional rigidity of a bar of any section equal to -j-l.
(§ 665) multiplied into the sum of its flexural rigidities (see below,
§ 679) in any two planes at right angles to one another through
its length. The true theory, as we have seen (§ 675), always
gives a torsional rigidity less than this. How great the deficiency
may be expected to be in cases in which the figure of the section
presents projecting angles, or considerable prominences (which may
be imagined from the hydrokinetic analogy we have given in § 675),
has been pointed out by M. de St. Venant, with the important
practical application, that strengthening ribs, or projections (see, for
instance, the fourth annexed diagram), such as are introduced in
engineering to give stiffness to beams, have the reverse of a good
effect when torsional rigidity or strength is an object, although they
are truly of great value in increasing the flexural rigidity, and giving
STATICS OF SOLIDS. AND FLUIDS.
259
strength to bear ordinary strains, which are always more or less
flexural. With remarkable ingenuity and mathematical skill he has
drawn beautiful illustrations of this important practical principle from
his algebraic and transcendental solutions. Thus for an equilateral
(2)
(3)
(4)
(l)
Square with curved
Square with acute
Star with four
(5)
Rectilineal
corners and hollow
angles and hollow
rounded points,
Equilateral
square.
sides.
sides.
being a curve of
the eighth degree.
triangle.
•84346.
•88326.
•8186.
•8666.
•7783.
•8276.
•5874.
•6745.
•60000.
•72552.
triangle, and for the rectilineal and three curvilineal squares shown
in the annexed diagram, he finds for the torsional rigidities the
values stated. The number immediately below the diagram indicates
in each case the fraction which the true torsional rigidity is of the
old fallacious estimate (§ 673); the latter being the product of the
rigidity of the substance into the moment of inertia of the cross
section round an axis perpendicular to its plane through its centre
of inertia. The second number indicates in each case the fraction
which the torsional rigidity is of that of a solid circular cylinder
of the same sectional area.
678. M. de St. Venant also calls attention to a conclusion from
his solutions which to many may be startling, that in his simpler cases
the places of greatest distortion are those points of the boundary
which are nearest to the axis of the twisted prism in each case, and
the places of least distortion those farthest from it. Thus in the
elliptic cylinder the substance is most strained at the ends of the smaller
principal diameter, and least at the ends of the greater. In the
equilateral triangular and square prisms there are longitudinal lines of
maximum strain through the middles of the sides. In the oblong
rectangular prism there are two lines of greater maximum strain
through the middles of the broader pair of sides, and two lines of less
maximum strain through the middles of the narrow sides. The
strain is, as we may judge from (§ 675) the hydrokinetic analogy,
excessively small, but not evanescent, in the projecting ribs of a prism
of the figure shown in (4) § 677. It is quite evanescent infinitely near
the angle, in the triangular and rectangular prisms, and in each other
case as (3) of § 677, in which there is a finite angle, whether acute
or obtuse, projecting outwards. This reminds us of a general
remark we have to make, although consideration of space may
17-
26o ABSTRACT DYNAMICS.
oblige us to leave it without formal proof. A solid of any elastic
substance, isotropic or aeolotropic, bounded by any surfaces pre-
senting projecting edges or angles, or re-entrant angles or edges,
however obtuse, cannot experience any finite stress or strain in the
neighbourhood oi 2. projecting 2iVi<^Q (trihedral, polyhedral, or conical);
in the neighbourhood of an edge, can only experience simple longi-
tudinal stress parallel to the neighbouring part of the edge; and
generally experiences infinite stress and strain in the neighbourhood
of a re-entrant edge or angle; when influenced by any distribution
of force, exclusive of surface tractions infinitely near the angles or
edges in question. An important application of the last part of
this statement is the practical rule, well known in mechanics, that
every re-entering edge or angle ought to be rounded to prevent risk
of rupture, in solid pieces designed to bear stress. An illustration
of these principles is afforded by the complete mathematical solution
of the torsion problem for prisms of fan-shaped sections, such as
the annexed figures. In the cases corresponding to figures (4), (5),
(6) below, the distortion at the centre of the circle vanishes in (4),
is finite and determinate in (5), and infinite in (6).
(i) (2) (3) (4) (5) (6)
679. Hence in a rod of isotropic substance the principal axes
of flexure (§ 609) coincide with the principal axes of inertia of the
area of the normal section ; and the corresponding flexural rigidities
are the moments of inertia of this area round these axes multi-
plied by Young's modulus. Analytical investigation leads to the
following results, due to St. Venant. Imagine the whole rod di-
vided, parallel to its length, into infinitesimal filaments (prisms when
the rod is straight). Each of these contracts or swells laterally with
sensibly the same freedom as if it were separated from the rest of
the substance, and becomes elongated or shortened in a straight line
to the same extent as it is really elongated or shortened in the circular
arc which it becomes in the bent rod. The distortion of the cross
section by which these changes of lateral dimensions are necessarily
accompanied is illustrated in the annexed diagram, in which either the
whole normal section of a rectangular beam, or a rectangular area in
the normal section of a beam of any figure, is represented in its strained
and unstrained figures, with the central point O common to the two.
The flexure is in planes perpendicular to YO Y^ , and concave upwards
(or towards X); G the centre of curvature, being in the direction
indicated, but too far to be included in the diagram. The straiglit
STATICS OF SOLIDS AND FLUIDS.
261
sides AC, BD, and all straight lines parallel to them, of the unstrained
rectanojular area become concentric arcs of circles concave in the
16^
A
F
c
T — 0 J-—
F'
3-J T^
V
\\\\i
ZAAit
7
1- \
\ — — V — ^ —
f\
/ T
^ \
— \ \ \
U
jjTn
/ ^
mtr
B
E^
~itztii
D
t
y-
B'
^^^'d
H
opposite direction, their centre of curvature, H, being for rods of
gelatinous substance, or of glass or metal, from 2 to 4 times as far
from O on one side as G is on the other. Thus the originally plane
sides AC, BD of a rectangular bar become anticlastic surfaces, of
curvatures - and — , in the two principal sections. A flat rectangular,
9 P
or a square, rod of India-rubber [for which a- amounts (§ 655) to very
nearly ^, and which is susceptible of very great amounts of strain
without utter loss of corresponding elastic action], exhibits this
phenomenon remarkably well,
680. The conditional limitation (§605) of the curvature to being very
small in comparison with that of a circle of radius equal to the greatest
diameter of the normal section (not obviously necessary, and indeed
not generally known to be necessary, we believe, when the greatest
diameter is perpendicular to the plane of curvature) now receives its
full explanation. For unless the hread^/i, AC, of the bar (or diameter
perpendicular to the plane of flexure) be very small in comparison
with the mean proportional between the radius, OH, and the thick-
ness, AB, the distances from OV to the corners A', C" would fall
short of the half thickness, OF, and the distances to B', D' would
exceed it by diflerences comparable with its own amount. This
would give rise to sensibly less and greater shortenings and stretchings
262 ABSTRACT DYNAMICS.
in the filaments towards the corners, and so vitiate the solution.
Unhappily mathematicians have not hitherto succeeded in solving,
possibly not even tried to solve, the beautiful problem thus presented
by the flexure of a broad very thin band (such as a watch spring) into
a circle of radius comparable with a third proportional to its thickness
and its breadth.
681. But, provided the radius of curvature of the flexure is not
only a large multiple of the greatest diameter, but also of a third
proportional to the diameters in and perpendicular to the plane of
flexure; then however great may be the ratio of the greatest diameter
to the least, the preceding solution is applicable; and it is remarkable
that the necessary distortion of the normal section (illustrated in the
diagram of § 679) does not sensibly impede the free lateral con-
tractions and expansions in the filaments, even in the case of a broad
thin lamina (whether of precisely rectangular section, or of unequal
thicknesses in different parts).
682. In our sections on hydrostatics, the problem of finding the
deformation produced in a spheroid of incompressible liquid by a
given disturbing force will be solved ; and then we shall consider the
application of the preceding methods to an elastic solid sphere in their
bearing on the theory of the tides and the rigidity of the earth. This
proposed application, however, reminds us of a general remark of
great practical importance, with which we shall leave elastic solids for
the present. Considering different elastic solids of similar substance
and similar shapes, we see that if by forces applied to them in any
way they are similarly strained, the surface tractions in or across
similarly situated elements of surface, whether of their boundaries
or of surfaces imagined as cutting through their substances, must be
equal, reckoned as usual per unit of area. Hence; the force across,
or in, any such surface, being resolved into components parallel to
any directions; the whole amounts of each such component for
similar surfaces of the different bodies are in proportion to the squares
of their lineal dimensions. Hence, if equilibrated similarly under the
action of gravity, or of their kinetic reactions (§ 230) against equal
accelerations (§ 32), the greater body would be more strained than the
less ; as the amounts of gravity or of kinetic reaction of similar
portions of them are as the mbes of their linear dimensions. Defi-
nitively, the strains at similarly situated points of the bodies will
be in simple proportion to their linear dimensions, and the displace-
ments will be as the squares of these lines, provided that there is no
strain in any part of any of them too great to allow the principle
of superposition to hold with sufficient exactness, and that no part is
turned through more than a very small angle relatively to any other
part. To illustrate by a single example, let us consider a uniform
long, thin, round rod held horizontally by its middle. Let its
substance be homogeneous, of density p, and Young's modulus, M'y
and let its length, /, be p times its diameter. Then (as the moment
of inertia of a circular area of radius r round a diameter is |7rr'*) the
STATICS OF SOLinS AND FLUIDS 263
flexural rigidity of the rod will (§ 679) be — tt ( — J . This gives us
for the curvature at the middle of the rod the elongation and con-
traction where greatest, that is, at the highest and lowest points of the
normal section through the middle point ; and the droop of the ends ;
the following expressions
M ' M' ^^ SM'
Thus, for a rod whose length is 200 times its diameter, if its substance
be iron or steel, for which p= 775, and M= 194 x 10^ grammes per
square centimetre, the maximum elongation and contraction (being
at the top and bottom of the middle section where it is held) are
each equal to '8 x 10"^ x /, and the droop of its ends 2 x io~^x /^
Thus a steel or iron wire, ten centimetres long, and half a millimetre
in diameter, held horizontally by its middle, would experience only
•000008 of maximum elongation and contraction, and only "002 of
a centimetre of droop in its ends : a round steel rod, of half a centi-
metre in diameter, and one metre long, would experience '00008 of
maximum elongation and contraction, and '2 of a centimetre of
droop : a round steel rod, of ten centimetres diameter, and twenty
metres long, must be of remarkable temper (see Properties of Matter)
to bear being held by the middle without taking a very sensible per-
manent set : and it is probable that no temper of steel is high enough
in a round shaft forty metres long, if only two decimetres in dia-
meter, to allow it to be held by its middle without either bending
it to some great angle, and beyond all appearance of elasticity, or
breaking it.
683. In passing from the dynamics of perfectly elastic solids to
abstract hydrodynamics, or the dynamics of perfect fluids, it is con-
venient and instructive to anticipate slightly some of the views as to
intermediate properties observed in real solids and fluids, which,
according to the general plan proposed (§ 402) for our work, will be
examined with more detail under Properties of Matter.
By induction from a great variety of observed phenomena, we are
compelled to conclude that no change of volume or of shape can be
produced in any kind of matter without dissipation of energy (§ 247);
so that if in any case .there is a return to the primitive configuration,
some amount (however small) of work is always required to com-
pensate the energy dissipated away, and restore the body to the same
physical and the same palpably kinetic condition as that in which it
was given. We have seen (§ 643), by anticipating something of
thermodynamic principles, how such dissipation is inevitable, even in
dealing with the absolutely perfect elasticity of volume presented by every
fluid, and possibly by some solids, as, for instance, homogeneous
crystals. But in metals, glass, porcelain, natural stones, wood, India-
rubber, homogeneous jelly, silk fibre, ivory, etc., a d\^\\xiQXfrictional
264 ABSTRACT DYNAMICS.
resistance^ against every change of shape is, as we shall see later,
under Properties of Matter^ demonstrated by many experiments, and
is found to depend on the speed with which the change of shape is
made. A very remarkable and obvious proof of frictional resistance
to change of shape in ordinary solids, is afforded by the gradual,
more or less rapid, subsidence of vibrations of elastic solids; mar-
vellously rapid in India-rubber, and even in homogeneous jelly; less
rapid in glass and metal springs, but still demonstrably, much more
rapid than can be accounted for by the resistance of the air. This
molecular friction in elastic solids may be properly called viscosity of
solids, because, as being an internal resistance to change of shape
depending on the rapidity of the change, it must be classed with
fluid molecular friction, which by general consent is called viscosity of
fluids. But, at the same time, we feel bound to remark that the word
viscosity, as used hitherto by the best writers, when solids or hetero-
geneous semisolid-semifluid masses are referred to, has not been
distinctly applied to molecular friction, especially not to the molecular
friction of a highly elastic solid within its hmits of high elasticity, but
has rather been employed to designate a property of slow, continual
yielding through very great, or altogether unlimited, extent of change
of shape, under the action of continued stress. It is in this sense
that Forbes, for instance, has used the word in stating that 'Viscous
Theory of Glacial Motion' which he demonstrated by his grand
observations on glaciers. As, however, he, and many other writers
after him, have used the words plasticity and plastic, both with refer-
ence to homogeneous solids (such as wax or pitch, even though also
brittle; soft metals; etc.), and to heterogeneous semisohd-semifluid
masses (as mud, moist earth, mortar, glacial ice, etc.), to designate
the property ^ common to all those cases, of experiencing, under
continued stress either quite continued and unlimited change of shape,
or gradually very great change* at a diminishing (asymptotic) rate
through infinite time ; and as the use of the term plasticity implies no
more than does viscosity, any physical theory or explanation of the
property, the word viscosity is without inconvenience left available
for the definition we have given of it above.
684. A perfect fluid, or (as we shall call it) a fluid, is an unrealizable
conception, like a rigid, or a smooth, body : it is defined as a body
incapable of resisting a change of shape : and therefore incapable of
experiencing distorting or tangential stress (§ 640). Hence its pres-
sure on any surface, whether of a solid or of a contiguous portion of
1 See Proceedings of the Royal Society, May 1865, * On the Viscosity and
Elasticity of Metals ' (W. Thomson).
2 Some confusion of ideas might have been avoided on the part of writers who
have professedly objected to Forbes' theory while really objecting only (and we
believe groundlessly) to his usage of the word viscosity, if they had paused to
consider that no one physical explanation can hold for those several cases; and
that Forbes' theory is merely the proof by observation that glaciers have the
property that mud (heterogeneous), mortar (heterogeneous), pitch (homogeneous).,
water (homogeneous), all have of changing shape indefinitely and continuously
under the action of continued stress.
STATICS OF SOLIDS AND FLUIDS. 265
the fluid, is at every point perpendicular to the surface. In equi-
librium, all common liquids and gaseous fluids fulfil the definition.
But there is finite resistance, of the nature of friction, opposing change
of shape at a finite rate; and, therefore, while a fluid is changing
shape, it exerts tangential force on every surface other than normal
planes of the stress (§ 635) required to keep this change of shape
going on. Hence; although the hydrostatical results, to which we
immediately proceed, are verified in practice; in treating of hydro-
kinetics, in a subsequent chapter, we shall be obliged to introduce the
consideration of fluid friction, except in cases where the circumstances
are such as to render its effects insensible.
685. With reference to a fluid the pressure at any point in any
direction is an expression used to denote the average pressure per unit
of area on a plane surface imagined as containing the point, and
perpendicular to the direction in question, when the area of that
surface is indefinitely diminished.
686. At any point in a fluid at rest the pressure is the same in
all directions: and, if no external forces act, the pressure is the same
at every point. For the proof of these and most of the following
propositions, we imagine, according to § 584, a definite portion of
the fluid to become solid, without changing its mass, form, or
dimensions.
Suppose the fluid to be contained in a closed vessel, the pressure
within depending on the pressure exerted on it by the vessel, and not
on any external force such as gravity.
687. The resultant of the fluid pressures on the elements of any
portion of a spherical surface must, like each of its components, pass
through the centre of the sphere. Hence, if we suppose (§ 584) a
portion of the fluid in the form of a plano-convex lens to be solidified,
the resultant pressure on the plane side must pass through the centre
of the sphere; and, therefore, being perpendicular to the plane, must
pass through the centre of the circular area. From this it is obvious
that the pressure is the same at all points of any plane in the fluid.
Hence the resultant pressure on any plane surface passes through
its centre of inertia.
Next, imagine a triangular prism of the fluid, with ends perpen-
dicular to its faces, to be solidified. The resultant pressures on its
ends act in the line joining the centres of inertia of their areas,
and are equal since the resultant pressures on the sides are in
directions perpendicular to this line. Hence the pressure is the same
in all parallel planes.
But the centres of inertia of the three faces, and the resultant
pressures applied there, lie in a triangular section parallel to the ends.
The pressures act at the middle points of the sides of this triangle,
and perpendicularly to them, so that their directions meet in a
point. And, as they are in equilibrium, they must be proportional
to the respective sides of the triangle; that is, to the breadths, or
areas, of the faces of the prism. Thus the resultant pressures on the
266 ABSTRACT DYNAMICS.
faces must be proportional to the areas of the faces, and therefore
the pressure is equal in any two planes which meet.
Collecting our results, we see that the pressure is the same at all
points, and in all directions, throughout the fluid mass.
688. Hence if a force be appHed at the centre of inertia of each
face of a polyhedron, with magnitude proportional to the area of
the face, the polyhedron will be in equiUbrium. For we may suppose
the polyhedron to be a solidified portion of the fluid. The resultant
pressure on each face will then be proportional to its area, and will
act at its centre of inertia; which, in this case, is the Centre of
Pressure.
689. Another proof of the equality of pressure throughout a mass
of fluid, uninfluenced by other external force than the pressure of the
containing vessel, is easily furnished by the energy criterion of equi-
librium, § 254; but, to avoid complication, we will consider the fluid
to be incompressible. Suppose a number of pistons fitted into
cylinders inserted in the sides of the closed vessel containing the fluid.
Then, if A be the area of one of these pistons, / the average pressure
on it, X the distance through which it is pressed, in or out; the energy
criterion is that no work shall be done on the whole, i.e. that
as much work being restored by the pistons which are forced out, as
is done by those forced in. Also, since the fluid is incompressible, it
must have gained as much space by forcing out some of the pistons
as it lost by the intrusion of the others. This gives
A^x^+A^x^ + ...=t{Ax) = o.
The last is the only condition to which x^^ x^^, etc., in the first equa*
tion, are subject; and therefore the first can only be satisfied if
A=A=/^a=etc.,
that is, if the pressure be the same on each piston. Upon this pro-
perty depends the action of Bramah's Hydrostatic Press.
If the fluid be compressible, the work expended in compressing it
from volume VXo F-BF, at mean pressure/, isJfSF.
If in this case we assume the pressure to be the same throughout,
we obtain a result consistent with the energy criterion.
The work done on the fluid is 5 {Apx)^ that is, in consequence of
the assumption, p% {Ax).
But this is equal to /8 F^
for, evidently, S {Ax) = BF.
690. When forces, such as gravity, act from external matter upon
the substance of the fluid, either in proportion to the density of its
own substance in its different parts, or in proportion to the density
of electricity, or of magnetic polarity, or of any other conceivable
accidental property of it, the pressure will still be the same in all
directions at any one point, but will now vary continuously from
point to point. For the preceding demonstration (§ 687) may still
STATICS OF SOLIDS AND FLUIDS 267
be applied by simply taking the dimensions of the prism small
enough; since the pressures are as the squares of its linear dimen-
sions, and the effects of the applied forces such as gravity, as the
cubes.
691. When forces act on the whole fluid, surfaces of equal pressure,
if they exist, must be at every point perpendicular to the direction of
the resultant force. For, any prism of the fluid so situated that the
whole pressures on its ends are equal must experience from the
applied forces no component in the direction of its length; and,
therefore, if the prism be so small that from point to point of it the
direction of the resultant of the applied forces does not vary sensibly,
this direction must be perpendicular to the length of the prism.
From this it follows that whatever be the physical origin, and the law,
of the system of forces acting on the fluid, and whether it be con-
servative or non-conservative, the fluid cannot be in equilibrium unless
the lines of force possess the geometrical property of being at right
angles to a series of surfaces.
692. Again, considering two surfaces of equal pressure infinitely
near one another, let the fluid between them be divided into columns
of equal transverse section, and having their lengths perpendicular to
the surfaces. The difference of pressure on the two ends being the
same for each column, the resultant applied forces on the fluid masses
composing them must be equal. Comparing this with § 506, we see
that if the applied forces constitute a conservative system, the density
of matter, or electricity, or whatever property of the substance they
depend on, must be equal throughout the layer under consideration.
This is the celebrated hydrostatic proposition that in a fluid at rest ,
surfaces of equal pressure are also surfaces of equal density and of equal
potential,
693. Hence when gravity is the only external force considered,
surfaces of equal pressure and equal density are (when of moderate
extent) horizontal planes. On this depends the action of levels,
siphons, barometers, etc.; also the separation of liquids of different
densities (which do not mix or combine chemically) into horizontal
strata, etc., etc. The free surface of a liquid is exposed to the pressure
of the atmosphere simply; and therefore, when in equilibrium, must
be a surface of equal pressure, and consequently level. In extensive
sheets of water, such as the American lakes, differences of atmo-
spheric pressure, even in moderately calm weather, often produce con-
siderable deviations from a truly level surface.
694. The rate of increase of pressure per unit of length in the
direction of the resultant force, is equal to the intensity of the force
reckoned per unit of volume of the fluid. Let F be the resultant
force per unit of volume in one of the columns of § 692; / and/'
the pressures at the ends of the column, / its length, S its section. We
have, for the equilibrium of the column,
{p'-p)S=SlF.
Hence the rate of increase of pressure per unit of length is F>
268 ABSTRACT DYNAMICS.
If the applied forces belong to a conservative system, for which
V and V are the values of the potential at the ends of the column,
we have (§ 504)
r-V=-lFp,
where p is the density of the fluid. This gives
p'-t=-p{V-V),
or dp= -pdV.
Hence in the case of gravity as the only impressed force the rate
of increase of pressure per unit of depth in the fluid is p, in gravitation
measure (usually employed in hydrostatics). In kinetic or absolute
measure (§ 189) it is gp.
If the fluid be a gas, such as air, and be kept at a constant tem-
perature, we have p = ^, where c denotes a constant, the reciprocal of
H^ the 'height of the homogeneous atmosphere,* defined (§ 695)
below. Hence, in a calm atmosphere of uniform temperature we
have
and from this, by integration,
where p^ is the pressure at any particular level (the sea-level, for
instance) where we choose to reckon the potential as zero.
When the differences of level considered are infinitely small in
comparison with the earth's radius, as we may practically regard them,
in measuring the heights of mountains, or of a balloon, by the baro-
meter, the force of gravity is constant, and therefore differences of
potential (force being reckoned in units of weight) are simply equal
to diflerences of level. Hence if x denote height of the level of
pressure/ above thatof/^, we have, in the preceding formulae, V=x,
and therefore
/=^„€--j that is,
695. If the air be at a constant temperature, the pressure
diminishes in geometrical progression as the height increases in
arithmetical progression. This theorem is due to Halley. Without
formal mathematics we see the truth of it by remarking that dif-
ferences of pressure are (§ 694) equal to differences of level multiplied
by the density of the fluid, or by the proper mean density when the
density differs sensibly between the two stations. But the density,
when the temperature is constant, varies in simple proportion to
the pressure, according to Boyle's law. Hence difl"erences of pres-
sure between pairs of stations diff"ermg equally in level are pro-
portional to the proper mean values of the whole pressure, which is
the well-known compound interest law. The rate of diminution of
pressure per unit of length upwards in proportion to the whole
pressure at any point, is of course equal to the reciprocal of the height
above that point that the atmosphere must have, if of constant
density, to give that pressure by its weight. The height thus defined
is commonly called 'the height of the homogeneous atmosphere,* a
STATICS OF SOLIDS AND FLUIDS 269
very convenient conventional expression. It is equal to the product
of the volume occupied by the unit mass of the gas at any pressure
into the value of tliat pressure reckoned per unit of area, in terms of
the weight of the unit of mass. If we denote it by H^ the expo-
nential expression of the law is
which agrees with the final formula of § 694.
The value of H for dry atmospheric air, at the freezing tem-
perature, according to Regnault, is, in the latitude of Paris, 799,020
centimetres, or 26,215 feet. Being inversely as the force of gravity
in different latitudes (§ 187), it is 798 533 centimetres, or 26,199 feet,
in the latitude of Edinburgh and Glasgow.
696. It is both necessary and sufficient for the equilibrium of an
incompressible fluid completely filling a rigid closed vessel, and
influenced only by a conservative system of forces, that its density be
uniform over every equipotential surface, that is to say, every surface
cutting the lines of force at right angles. If, however, the boundary,
or any part of the boundary, of the fluid mass considered, be not
rigid ; whether it be of flexible solid matter (as a membrane, or a thin
sheet of elastic solid), or whether it be a mere geometrical boundary,
on the other side of which there is another fluid, or 7iothing [a case
which, without believing in vacuum as a reality, we may admit in
abstract dynamics (§ 391)], a farther condition is necessary to secure
that the pressure from without shall fulfil the hydrostatic equation
at every point of the boundary. In the case of a bounding membrane,
this condition must be fulfilled either through pressure artificially
applied from without, or through the interior elastic forces of the
matter of the membrane. In the case of another fluid of different
density touching it on the other side of the boundary, all round or
over some part of it, with no separating membrane, the condition
of equilibrium of a heterogeneous fluid is to be fulfilled relatively
to the whole fluid mass made up of the two; which shows that at the
boundary the pressure must be constant and equal to that of the fluid
on the other side. Thus water, oil, mercury, or any other liquid, in
an open vessel, with its free surface exposed to the air, requires for
equilibrium simply that this surface be level.
697. Recurring to the consideration of a finite mass of fluid
completely filling a rigid closed vessel, we see, from what precedes,
that, if homogeneous and incompressible, it cannot be disturbed from
equilibrium by any conservative system of forces; but we do not
require the analytical investigation to prove this, as we should have
'the perpetual motion' if it were denied, which would violate the
hypothesis that the system of forces is conservative. On the other
hand, a non-conservative system of forces cannot, under any circum-
stances, equilibrate a fluid which is either uniform in density through-
out, or of homogeneous substance, rendered heterogeneous in density
only through difference of pressure. But if the forces, though not
270 ABSTRACT DYNAMICS.
conservative, be such that through every point of the space occupied
by the fluid a surface may be drawn which shall cut at right angles
all the lines of force it meets, a heterogeneous fluid will rest in
equilibrium under their influence, provided (§ 692) its density, from
point to point of every one of these orthogonal surfaces, varies in-
versely as the product of the resultant force into the thickness of
the infinitely thin layer of space between that surface and another of
the orthogonal surfaces infinitely near it on either side. (Compare
§ 506).
698. If we imagine all the fluid to become rigid except an infinitely
thin closed tubular portion lying in a surface of equal density, and if
the fluid in this tubular circuit be moved any length along the tube
and left at rest, it will remain in equilibrium in the new position,
all positions of it in the tube being indifferent because of its homo-
geneousness. Hence the work (positive or negative) done by the
force {X, V, Z) on any portion of the fluid in any displacement
along the tube is balanced by the work (negative or positive) done on
the remainder of the fluid in the tube. Hence a single particle, acted
on only by X, K, Z, while moving round the circuit, that is moving
along any closed curve on a surface of equal density, has, at the end
of one complete circuit, done just as much work against the force in
some parts of its course, as the forces have done on it in the re-
mainder of the circuit.
699. The following imaginary example, and its realization in a
subsequent section (§ 701), show a curiously interesting practical
application of the theory of fluid equilibrium under extraordinary
circumstances, generally regarded as a merely abstract analytical
tlieory, practically useless and quite unnatural, 'because forces in
nature follow the conservative law.'
700. Let the lines of force be circles, with their centres all in one
line, and their planes perpendicular to it. They are cut at right
angles by planes through this axis ; and therefore a fluid may be in
equilibrium under such a system of forces. The system will not be
conservative if the intensity of the force be according to any other law
than inverse proportionality to distance from this axial line; and the
fluid, to be in equilibrium, must be heterogeneous, and be so dis-
tributed as to vary in density from point to point of every plane
through the axis, inversely as the product of the force into the
distance from the axis. But from one such plane to another it may
be either uniform in density, or may vary arbitrarily. To particularize
farther, we may suppose the force to be in direct simple proportion
to the distance from the axis. Then the fluid will be in equilibrium
if its density varies from point to point of every plane through the
axis, inversely as the square of that distance. If we still farther
particularize by making the force uniform all round each circular line
of force, the distribution of force becomes precisely that of the kinetic
reactions of the parts of a rigid body against accelerated rotation.
The fluid pressure will (§ 691) be equal over each plane through the
STATICS OF SOLIDS AND FLUIDS. 271
axis. And in one such plane, which we may imagine carried round
the axis in the direction of the force, the fluid pressure will increase in
simple proportion to the angle at a rate per unit angle (§55) equal to
the product of the density at unit distance into the force at unit distance.
Hence it must be remarked, that if any closed line (or circuit) can be
drawn round the axis, without leaving the fluid, there cannot be
equilibrium without a firm partition cutting every such circuit, and
maintaining the diflerence of pressures on the two sides of it, corre-
sponding to the angle 2ir. Thus, if the
axis pass through the fluid in any part,
there must be a partition extending from
this part of the axis continuously to the
outer bounding surface of the fluid. Or if
the bounding surface of the whole fluid be
annular (like a hollow anchor-ring, or of
any irregular shape), in other words, if the
fluid fills a tubular circuit; and the axis
(A) pass through the aperture of the ring
(without passing into the fluid); there must be a firm partition (CD)
extending somewhere continuously across the channel, or passage, or
tube, to stop the circulation of the fluid round it; otherwise there
could not be equilibrium with the supposed forces in action. If we
further suppose the density of the fluid to be uniform round each of
the circular lines of force in the system we have so far considered (so
that the density shall be equal over every circular cylinder having the
line of their centres for its axis, and shall vary from one such
cylindrical surface to another, inversely as the squares of their radii),
we may, without disturbing the equiHbrium, impose any conservative
system of force in lines perpendicular to the axis; that is (§ 506), any
system of force in this direction, with intensity varying as some
function of the distance. If this function be the simple distance, the
superimposed system of force agrees precisely with the reactions
against curvature, that is to say, the centrifugal forces, of the parts of
a rotating rigid body.
701. Thus we arrive at the remarkable conclusion, that if a rigid
closed box be completely filled with incompressible heterogeneous
fluid, of density varying inversely as the square of the distance from
a certain line, and if the box be movable round this line as a fixed
axis, and be urged in any way by forces applied to its outside, the
fluid will remain in equilibrium relatively to the box; that is to say,
will move round with the box as if the whole were one rigid body,
and will come to rest with the box if the box be brought again to
rest: provided always the preceding condition as to partitions be
fulfilled if the axis pass through the fluid, or be surrounded by
continuous lines of fluid. For, in starting from rest, if the fluid
moves like a rigid solid, we have reactions against acceleration,
tangential to the circles of motion, and equal in amount to wr per
unit of mass of the fluid at distance r from the axis, w being the rate
272 ABSTRACT DYNAMICS.
of acceleration (§ 57) of the angular velocity; and (see Vol. II.) we
have, in the direction perpendicular to the axis outwards, reaction
against curvature of path, that is to say, 'centrifugal force,' equal to
o)V per unit of mass of the fluid. Hence the equilibrium which we
have demonstrated in the preceding section, for the fluid supposed
at rest, and arbitrarily influenced by two systems of force (the circular
non-conservative and the radial conservative system) agreeing in law
with these forces of kinetic reaction, proves for us now the D'Alem-
bert (§ 230) equilibrium condition for the motion of the whole fluid as
of a rigid body experiencing accelerated rotation: that is to say,
shows that this kind of motion fulfils for the actual circumstances the
laws of motion, and, therefore, that it is the motion actually taken by
the fluid.
702. In § 688 we considered the resultant pressure on a plane
surface, when the pressure is uniform. We may now consider
(briefly) the resultant pressure on a plane area when the pressure
varies from point to point, confining our attention to a case of
great importance ;^that in which gravity is the only applied force,
and the fluid is a nearly incompressible liquid such as water. In this
case the determination of the position of the Centre of Pressure is
very simple ; and the whole pressure is the same as if the plane area
were turned about its centre of inertia into a horizonal position.
The pressure at any point at a depth z in the liquid may be ex-
pressed by
where p is the (constant) density of the liquid, and /^ the (atmo-
spheric) pressure at the free surface, reckoned in units of weight per
unit of area.
Let the axis of x be taken as the intersection of the plane of the
immersed plate with the free surface of the liquid, and that of y
perpendicular to it and in the plane of the plate. Let a be the
inclination of the plate to the vertical. Let also A be the area of the
portion of the plate considered, and x, y, the co-ordinates of its centre
of inertia.
Then the whole pressure is
jjj^dxdy = // (/„ + py cos a) dxdy
= Ap^ 4 ^p^ cos a.
The moment of the pressure about the axis of Jt: is
jjpydxdy = Ap^y + Ak^p cos a,
k being the radius of gyration of the plane area about the axis of x.
For the moment about y we have
jjpxdxdy = Ap^x + p cos a jjxydxdy.
The first terms of. these three expressions merely give us again the
results of § 688; we may therefore omit them. This will be equi-
valent to introducing a stratum of additional liquid above the free
surface such as to produce an equivalent to the atmospheric pressure.
STATICS OF SOLIDS AND FLUIDS. 273
If the origin be now shifted to the upper surface of this stratum we
have
Pressure = Apy cos a,
Moment about Ox = Ak^p cos a,
Distance of centre of pressure from axis oi x = —.
But if /', be the radius of gyration of the plane area about a horizontal
axis in its plane, and passing through jts centre of inertia, we have
Hence the distance, measured parallel to the axis of j, of the centre
of pressure from the centre of inertia is
and, as we might expect, diminishes as the plane area is more and
more submerged. If the plane area be turned about the line through
its centre of inertia parallel to the axis of x^ this distance varies as
the cosine of its inclination to the vertical; supposing, of course, that
by the rotation neither more nor less of the plane area is submerged.
703. A body, wholly 6t partially immersed in any fluid influenced
by gravity, loses, through fluid pressure, in apparent weight an amount
equal to the weight of the fluid displaced. For if the body were
removed, and its place filled with fluid homogeneous with the sur-
rounding fluid, there would be equilibrium, even if this fluid be sup-
posed to become rigid. And the resultant of the fluid pressure upon
it is therefore a single force equal to its weight, and in the vertical
line through its centre of gravity. But the fluid pressure on the
originally immersed body was the same all over as on the solidified
portion of fluid by which for a moment we have imagined it replaced,
and therefore must have the same resultant. This proposition is of
great use in Hydrometry, the determination of specific gravity, etc.,
etc.
704. The following lemma, while in itself interesting, is of great
use in enabling us to simplify the succeeding investigations regarding
the stability of equilibrium of floating bodies : —
Let a homogeneous solid, the weight of unit of volume of which
we suppose to be unity, be cut by a horizontal plane in XYX'Y.
Let O be the centre of inertia, 1
and let XX\ YY' be the principal 1'
axes, of this area.
Let there be a second plane
section of the solid, through YY\
inclined to the first at an infinitely X'\
small angle, B. Then (i) the
volumes of the two wedges cut
from the solid by these sections
are equal; (2) their centres of
inertia lie in one plane perpen-
18
2 74 ABSTRACT DYNAMICS.
dicular X.o YY \ and (3) the moment of the weight of each of these,
round YY^ is equal to the moment of inertia about it of the corre-
sponding portion of the area multipHed by Q.
Take OX^ (9 F as axes, and let Q be the angle of the wedge : the
thickness of the wedge at any point P, (^, y)^ is Ox, and the volume
of a right prismatic portion whose base is the elementary area dxdy
at P is Bxdxdy.
Now let [ ] and ( ) be employed to distinguish integrations extended
over the portions of area to the right and left of the axis of j re-
spectively, while integrals over the whole area have no such distin-
guishing mark. Let v and v' be the volumes of the wedges ; (x, 7),
(^', y') the co-ordinates of their centres of inertia. Then
v-B\ ffxdxdy]
-v' = 0{jjxdxdy),
whence v-v' -0 jjxdxdy = o since O is the centre of inertia. Hence
v = v', which is (i).
Again, taking moments about XX' ^
vy = 0 [ffxydxdy] ,
and — vy = 6 (ffxydxdy).
Hence vy - v'y' = 0 ffxydxdy.
But for a principal axis '%xydm vanishes. Hence ly — v'y' -o^
whence, since v - v'^ we have
y-y'-) which proves (2).
And (3) is merely a statement in words of the obvious equation
[ffx.xOdxdy] = 6 [ffx^.dxdy].
705. If a positive amount of work is required to produce any
possible infinitely small displacement of a body from a position of
equilibrium, the equilibrium in this position is stable (§ 256). To
apply this test to the case of a floating body, we may remark, first,
that any possible infinitely small displacement may (§§ 30, 106) be
conveniently regarded as compounded of two horizontal displacements
in lines at right angles to one another, one vertical displacement, and
three rotations round rectangular axes through any chosen point. If
one of these axes be vertical, then three of the component displace-
ments, viz. the two horizontal displacements and the rotation about
the vertical axis, require no work (positive or negative), and therefore,
so far as they are concerned, the equilibrium is essentially neutral.
But so far as the other three modes of displacement are concerned,
the equilibrium may be positively stable, or may be unstable, or may
be neutral, according to the fulfilment of conditions which we now
proceed to investigate.
706. If, first, a simple vertical displacement, downwards, let us
suppose, be made, the work is done against an increasing resultant
of upward fluid pressure, and is of course equal to the mean increase
of this force multiplied by the whole space. If this space be denoted
by z, the area of the plane of flotation by A, and the weight of unit
bulk of the liquid by 7e>, the increased bulk of immersion is clearly Az,
STATICS OF SOLIDS AND FLUIDS.
275
and therefore the increase of the resultant of fluid pressure is wAz,
and is in a Hne vertically upward through the centre of gravity of A.
The mean force against which the work is done is therefore \7vAz,
as this is a case in which work is done against a force increasing
from zero in simple proportion to the space. Hence the work done
is ^wAz^. We see, therefore, that so far as vertical displacements
alone are concerned, the equilibrium is necessarily stable, unless the
body is wholly immersed, when the area of the plane of flotation
vanishes, and the equilibrium is neutral.
707. The lemma of § 704 suggests that we should take, as the
two horizontal axes of rotation, the principal axes of the plane of
flotation. Considering then rotation through an infinitely small angle
6 round one of these, let G and E be the displaced centres of gravity
of the solid, and of the portion of its volume which was immersed
when it was floating in equilibrium, and G', FJ the positions which
they then had; all projected on the plane of the diagram which we
suppose to be through / the centre of inertia of the plane of flotation.
The resultant action of gravity on the displaced body is W, its weight,
acting downwards through G\ and that of the fluid pressure on it is
W upwards through E corrected by the amount (upwards) due to the
additional immersion of the wedge AIA', and the amount (down-
wards) due to the extruded wedge B'IB. Hence the whole action of
18—2
2 76 ABSTRACT DYNAMICS.
gravity and fluid pressure on the displaced body is the couple of
forces up and down in verticals through G and E^ and the correction
due to the wedges. This correction consists of a force vertically
upwards through the centre of gravity of A' I A, and downwards
through that of BIB'. These forces are equal [§ 704 (i)], and
therefore constitute a couple which [704 (2)] has the axis of the
displacement for its axis, and which [§ 704 (3)] has its moment equal
to Qwk~A if A be the area of the plane of flotation, and k its radius
of gyration (§ 235) round the principal axis in question. But since
GE^ which was vertical {G'E') in the position of equilibrium, is
incHned at the infinitely small angle 6 to the vertical in the displaced
body, the couple of forces W in the verticals through G and E has
for moment WhS, if h denote GE] and is in a plane perpendicular
to the axis, and in the direction tending to increase the displacement,
when G is above E. Hence the resultant action of gravity and fluid
pressure on the displaced body is a couple whose moment is
{wAk^ - Wh)e, or w {Ak^ - Vh)e,
if V be the volume immersed. It follows that when Ak~> Vh the
equilibrium is stable, so far as this displacement alone is concerned.
Also, since the couple worked against in producing the displace-
ment increases from zero in simple proportion to the angle of dis-
placement, its mean value is half the above; and therefore the whole
amount of work done is equal to
Iw^Ak""- Vh)e\
708. If now we consider a displacement compounded of a vertical
(downwards) displacement z, and rotations through infinitely small
angles 6, 6' round the two horizontal principal axes of the plane of
flotation, we see (§§ 706, 70^) that the work required to produce it is
equal to
\7v [Az' + {Ak' - Vh) 0' + {Ak" -^ Vh) $"1
and we conclude that, for complete stability with reference to all pos-
sible displacements of this kind, it is necessary and sufficient that
^ Ak' ^ Ak"
n < -p.- , and < — ^- .
709. When the displacement is about any axis through the centre
of inertia of the plane of flotation, the resultant of fluid pressures is
equal to the weight of the body; but it is only when the axis is a
principal axis of the plane of flotation that this resultant is in the
plane of displacement. In such a case the point of intersection of
the resultant with the Hne originally vertical, and through the centre
of gravity of the body, is called the Metacenti'e. And it is obvious,
from the above investigation, that for either of these planes of dis-
placement the condition of stable equilibrium is that the metacentre
shall be above the centre of gravity.
710. We shall conclude with the consideration of one case of the
STATICS OF SOLIDS AND FLUIDS.
277
equilibrium of a revolving mass of fluid subject only to the gravitation
of its parts, which admits of a very simple synthetical solution, without
any restriction to approximate sphericity; and for which the following
remarkable theorem was discovered by Newton and Maclaurin : —
711. An oblate ellipsoid of revolution, of any given eccentricity, is
a figure of equilibrium of a mass of homogeneous incompressible
fluid, rotating about an axis with determinate angular velocity, and
subject to no forces but those of gravitation among its parts.
The angular velocity for a given eccentricity is independent of the
bulk of the fluid, and proportional to the square root of its density.
712. The proof of this proposition is easily obtained from the
results already deduced with respect to the attraction of an ellipsoid
and the properties of the free surface of a fluid.
We know, §538, that li AFB he. a meridian section of a homo-
geneous oblate spheroid, A C the polar axis, CF an equatorial radius,
and /'any point on the surface, the attraction of the spheroid may be
resolved into two parts;
one, Fp, perpendicular to
the polar axis, and vary-
ing as the ordinate FM;
the other, Fs, parallel to
the polar axis, and vary-
ing as FN. These com-
ponents are not equal
when MF and FN are
equal, else the resultant
attraction at all points in
the surface would pass
through C; whereas we
know that it is in some
such direction as Ff, cutting the radius BC between B and C, but at
a point nearer to C than n the foot of the normal at F. Let then
Fp=^a.FM,
and Fs = p.FN,
where a and ^ are known constants, depending merely on the density
(p), and eccentricity (^), of the spheroid.
Also, we know by geometry that Nn = (t - /) CN.
Hence; to find the magnitude of a force Fq perpendicular to the
axis of the spheroid, which, when compounded with the attraction,
will bring the resultant force into the normal Fn : make pr = Fq, and
we must have
PFp
Fr _ Nn
Fs ~ FN
= (i
O
aFs
Hence
Fr
e^)^Fp,
2)8 ABSTRACT DYNAMICS.
Pp-Pq = (^-e)^^Pp,
or
^^ = {i-(i-^)f}^/
Now if the spheroid were to rotate with angular velocity <o about AC,
the centrifugal force, §§ 39, 42, 225, would be in the direction Pq,
and would amount to
Hence, if we make
aj' = a-(i-^^)/?,
the whole force on P, that is, the resultant of the attraction and
centrifugal force, will be in the direction of the normal to the sur-
face, which is the condition for the free surface of a mass of fluid in
equilibrium.
Now, (§522 of our larger work)
l^^-e' . _, I- A
_ /I Ji-e' . _, \
/3 = 47rp\^~, ^^- sm ej,
XT 2 f(3 - 2^') Ji--e^ . _, I - /) / V
Hence ui=27rp r'? — — {^- sm V- 3 — ^ V . (i)
This determines the angular velocity, and proves it to be proportional
to Jp.
713. If, after Laplace, \ve introduce instead of ^ a quantity «
defined by the equation
I
-2 >
i-e- =
I + e'
\ (')
or g = — . = tan (sin V), |
n/i-^ J
the expression (i) for w^ is much simplified, and
When e, and therefore also c, is small, this formula is most easily
calculated from
^^ = A^'-A'' + etc. (4)
of which the first term is sufficient when we deal with spheroids so
little oblate as the earth.
The following table has been calculated by means of these simpli-
fied formulae. The last figure in each of the four last columns is
given to the nearest unit. The two last columns will be explained a
few sections later : —
STATICS OF SOLIDS AND FLUIDS.
279
e. i
I
1
— when 0 = 3-68x10"'^.
(i+e--^)^*^'.
j
1
e
2'irp'
w '^ ^
lirp
O'l
9'95o
0-0027
79,966
0*0027
•2 1
4-899 1
•OTO7
39»397
•Olio
•3
3-180
•0243
26,495
•0258
•4
2-291
•0436
19,780
•0490
•5
1-732
•0690
15.730
•0836
•6
i'333
•1007
13,022
•1356
•7
1-020
•1387
11,096
•2172
•8
0-750
•1816
9,697
•3588
•9
•4843
•2203 .
8,804
•6665
•91
•4556
•2225
8,759
•7198
•92
•4260
•2241
8,729
•7813
•93
•3952
•2247
8,718
•8533
•94
•3629
•2239
8,732
•9393
•95
•3287
•2213
8,783
I -045
1 -96
•2917
•2160
8,891
1-179
•97
•2506
•2063
9,098
i'359
•98
•2030
•1890
9,504
1-627
•99
•1425
'1551
10,490
2-113
i-oo
0-0000
O'OOOO
00
00
From this we see that the value of — • increases gradually from
27rp
zero to a maximum as the eccentricity e rises from zero to about 0-93,
and then (more quickly) falls to zero as the eccentricity rises from
o*93 to unity. The values of the other quantities corresponding to
this maximum are given in the table.
714. If the angular velocity exceed the value calculated from
^— = 0*2247,
27rp
(5)
when for p is substituted the density of the liquid, equilibrium Is im-
possible in the form of an ellipsoid of revolution. If the angular
velocity fall short of this limit there are always two ellipsoids of
revolution which satisfy the conditions of equilibrium. In one of
these the eccentricity is greater than 0*93, in the other less.
715. It may be useful, for special apphcations, to indicate briefly
how p is measured in these formulae. In the definitions of §§ 476,
477, on which the attraction formulae are based, unit mass is defined
as exerting unit force on unit mass at unit distance; and unit volume-
density is that of a body which has unit mass in unit volume. Hence,
with the foot as our linear unit, we have for the earth's attraction on
a particle of unit mass at its surface
^^<jR^
R'
rR
32
28o ABSTRA CT D YNAMICS.
where R Is the radius of the earth (supposed spherical) in feet, and
o- its mean density, expressed in terms of the unit just defined.
Taking 20,900,000 feet as the value of ^, we have
c = o'oooooo368 = 3*68 x lo"'^. (6)
As the mean density of the earth is somewhere about 5*5 times that
of water, the density of water in terms of our present unit is
^ 10- = 67x10-
5-5 '
716. The fourth column of the table above gives the time of rota-
tion in seconds, corresponding to each value of the eccentricity, p
being assumed equal to the mean density of the earth. For a mass
of water these numbers must be multiplied by sJs'S'} ^s the time of
rotation to give the same figure is inversely as the square root of the
density.
For a homogeneous liquid niass, of the earth's mean density,
rotating in 23^^ 46"^ 4^ we find ^=0*093, which corresponds to an
ellipticity of about ^io-
717. An interesting form of this problem, also discussed by Laplace,
is that in which the moment of momentum and the mass of the fluid
are given, not the angular velocity; and it is required to find what is
the eccentricity of the corresponding ellipsoid of revolution, the result
proving that there can be but one.
It is evident that a mass of any ordinary liquid (not 2l perfect fluids
§ 684), if left to itself in any state of motion, must preserve unchanged
its moment of momentum, § 202. But the viscosity, or internal
friction, § 684, will, if the mass remain continuous, ultimately destroy
all relative motion among its parts ; so that it will ultimately rotate as
a rigid solid. If the final form be an ellipsoid of revolution, we can
easily show that there is a single definite value of its eccentricity.
But, as it has not yet been discovered whether there is any other
form consistent ^\\ki stable equilibrium, we do not know that the mass
will necessarily assume the form of this particular ellipsoid. Nor in
fact do we know whether even the ellipsoid of rotation may not become
an unstable form if the moment of momentum exceed some limit de-
pending on the mass of the fluid. We shall return to this subject in
Vol. II., as it afl'ords an excellent example of that difficult and delicate
question Kinetic Stability^ § 300.
If we call a the equatorial semi-axis of the ellipsoid, e its eccen-
tricity, and to its angular velocity of rotation, the giz>€n quantities are
the mass M-^-rrpa^ J 1 —e^,
and the moment of momentum
A = -^^Trpoia' J I - e\
I'hese equations, along with (3), determine the three quantities, a, r,
and (o.
Eliminating a between the two just written, and expressing e as
before in terms of €, we have
STATICS OF SOLIDS AND FLUIDS. 281
This gives
O)^ k
^""P (l + 6^)^
where y^ is a gk'en multiple of ph Substituting in 771 (2) we have
^ = (1+ ^) ^^ (^^-^tan-U - pj
Now the last column of the table in § 713 shows that the value of this
function of € (which vanishes with e) continually increases with e, and
becomes infinite when c is infinite. Hence there is always one, and
only one, value of c, and therefore of e, which satisfies the conditions
of the problem.
718. All the above results might without much difficulty have been
obtained analytically, by the discussion of the equations; but we have
preferred, for once, to show by an actual case that numerical calcula-
tion may sometimes be of very great use.
719. No one seems yet to have attempted to solve the general
problem of finding all the forms of equilibrium which a mass of
homogeneous incompressible fluid rotating with uniform angular
velocity may assume. Unless the velocity be so small that the figure
differs but little from a sphere, the problem presents difficulties of an
exceedingly formidable nature. It is therefore of some importance
to know that we can by a synthetical process show that another form,
besides that of the elHpsoid of revolution, may be compatible with
equilibrii^m; vi?;. an ellipsoid with three unequal ajces, of which the
least is the axis of rotation. This curious theorem was discovered by
Jacobi in 1834, and seems, simple as it is, to have been enunciated
by him as a challenge to the French mathematicians \ For the proot
we must refer to our larger work.
^ See a Paper by Liouville, Journal de VAcqle Poly technique^ cahier xxiii., foot-
note to p. 290.
APPENDIX.
KINETICS.
{a) In the case of the Simple Pendidum, a heavy particle is sus-
pended from a point by a hght inextensible string. If we suppose it
to be drawn aside from the vertical position of equilibrium and
allowed to fall, it will oscillate in one plane about its lowest position.
When the string has an inclination Q to the vertical, the weight mg of
the particle may be resolved into mg cos B which is balanced by the
tension of the string, and mg sin 6 in the direction of the tangent to
the path. If / be the length of the string, the distance (along the arc)
from the position of equilibrium is 10.
Now if the angle of oscillation be small (not above 3° or 4° say), the
sine and the angle are nearly equal to each other. Hence the acce-
leration of the motion (which is rigorously g sin 0) may be written gB.
Hence we have a case of motion in which the acceleration is propor-
tional to the distance from a point in the path, that is, by § 74, Simple
Harmonic Motion, The square of the angular velocity in the cor-
responding circular motion is -^. — , = - , and the period of the
displacement /
harmonic motion is therefore 27r* /-. In the case of the pendulum,
the time of an oscillation from side to side of the vertical is usually
taken — and is therefore ^a / - •
((^) Thus the times of vibration of different pendulums are as the
square roots of their lengths, for any arcs of vibration, provided only
these be small.
Also the times of vibration of the same pendulum at different
places are inversely as the square roots of the apparent force of gravity
on a unit mass at these places.
(c) It was found experimentally by Newton that pendulums of the
same length vibrate in equal times at the same place whatever be the
material of which their bobs are formed. This would evidently not
be the case unless the weight were in every case proportional to the
amount of matter in the bob.
APPENDIX. 283
{i{) If the simple pendulum be slightly disturbed in any way from
its position of equilibrium, it will in general describe very nearly an
eUipse about its lowest position as centre. This is easily seen from
§82.
ii) If the arc of vibration be considerable, the motion will not be
simple harmonic, and the time of vibration will be greater than that
above stated; since the acceleration being as the sine of the dis-
placement, is in less and less ratio to the displacement as the latter is
greater.
In this case, the motion for any disturbance is, for one revolution,
approximately elliptic as before; but the ellipse slowly turns round
the vertical, in the direction in which the bob moves.
(/) The bob may, however, be so projected as to revolve uniformly
in a horizontal circle, in which case the apparatus is called a Conical
Penduhun. Here we have /sin B for the radius of the circle, and the
•force in the direction of the radius is T^sin ^, where J'is the tension of
the string. T'cos B balances mg — and thus the force in the radius of
the circle is w^tan 0. The square of the angular velocity in the circle
is therefore t-^^-tt, and the time of revolution 27r . / : or
/cos 6 V g '
where /i is the height of the point of suspension above the
plane of the circle. Thus all conical pendulums with the same height
revolve in the same time.
(g) A rigid mass oscillating about a horizontal axis, under the
action of gravity, constitutes what is called a Compound Pmdulwn.
When in the course of its motion the body is inclined at any angle
Q to the position in which it hangs, when in equilibrium, it experiences
from gravity, and the resistance of the supports of its axis, a couple,
which is easily seen to be equal to
gWh sin e,
where Wis the mass and h the distance of its centre of gravity fronl
the axis. This couple produces (§§ 232, 235) acceleration of angular
velocity, calculated by dividing the moment of the couple by the
moment of inertia of the body. Hence, if / denote the moment of
inertia about the supporting axis, the angular acceleration is equal to
gWsvcv 6
I '
Its motion is, therefore, identical (§ {a)) with that of the simple pen-
dulum of length equal to 77^ .
If a rigid body be supported about an axis, which either passes
very nearly through the centre of gravity, or is at a very great dis-
tance from this point, the length of the equivalent simple pendulum
will be very great : and it is clear that some particular distance for
the point of support from the centre of gravity will render the length
284 APPENDIX,
of the corresponding simple pendulum, and, therefore, the time of
vibration, least possible.
To investigate these circumstances for all axes parallel to a given
line, through the centre of gravity, let k be the radius of gyration
round this line, we have (§ 198),
and, therefore, if / be the length of the isochronous simple pendulum,
k^ 4- k' _ {h-kY + 2hk _ ^^^ (/i-ky
~ h ~ h h '
The second term of the last of these forms vanishes when h = k, and
is positive for all other values of h. The smallest value of / is,
therefore, 2k, and this, the shortest length of the isochronous simple
pendulum, is realized when the axis of support is at the distance k
from the centre of inertia.
To find at what distance //, from the centre of inertia the axis must
be fixed to produce a pendulum isochronous with the simple pen-
dulum, of given length /, we have the quadratic equation
For the solution to be possible we have seen that / must be greater
than, or at least equal to, 2k. If /= 2k, the roots of this equation are
equal, k being their common value. For any value of / greater than
2k, the equation has two real roots whose sum is equal to /, and pro-
duct equal to k^ : hence, for any distance from the centre of inertia
less than k, another distance greater than k, which is a third propor-
tional to it and k, gives the same time of vibration ; and the length of
the simple pendulum corresponding to either case, is equal to the sum
of the distances of the two axes from the centre of inertia. This sum
is equal to the distance between them if the two axes are in one
plane, through the; centre of inertia, and on opposite sides of this
point; and, therefore, for axes thus placed, g,nd not equidistant from
the centre of inertia, if the times of oscillation of the body when
successively supported upon them are found to be equal, it may be
inferred that the distance between them is equal to the length of
the isochronous simple pendulum. As a simple pendulum exists only
in theory, this proposition was taken advantage of by Kater for the
practical determination of the force of gravity at any station.
{h) A uniformly heavy and perfectly flexible cord, placed in the in-
terior of a smooth tube in the form of any plane curve, and subject to
no external forces, 7vill exert no pressure on the tube if it have every-
where the same tension, and move with a certain definiie velocity.
For, as in § 592, the statical pressure due to the curvature of the
Q
rope per unit of length is J'- (where <j is the length of the arc AB
in that figure) directed inwards to the centre of curvature. Now, the
element a-, whose mass is nitr, is moving in a curve whose curvature is
-with velocity v (suppose). The requisite force is - =mv'0\
APPENDIX. 285
Q
and for unit of length mt^-. Hence if T= mv^ the theorem is true.
If we suppose a portion of the tube to be straight, and the whole to
be moving with velocity v parallel to this line, and against the motion
of the cord, we shall have the straight part of the cord reduced to
rest, and an undulation, of any^ but unvarying^ form and dimensions,
.It
running along it with the linear velocity . / — .
Suppose the cord stretched by an appended mass of W'^pounds, and
suppose its length / feet and its own mass w pounds. Then T=^ IVg,
hn = Wy and the velocity of the undulation is
/
IV/e-
— ^ feet per second.
(J) When a?i incompressible liquid escapes from an orifice^ the velocity
is the same as ivould be acquired by falling from the free surface to the
level of the orifice.
For, as we may neglect (provided the vessel is large compared with
the orifice) the kinetic energy of the bulk of the liquid; the kinetic
energy of the escaping liquid is due to the loss of potential energy
of the whole by the depression of the free surface. Thus the pro-
position at once.
{k) The small oscillations of a liquid in a U tube follow the
harmonic law.
The tube being of uniform section S^ a depression of level, x,
from the mean, on one side, leads to a rise, x^ on the other; and if
the whole column of fluid be of length 2a^ we have the mass 2aSp
disturbed through a space x^ and acted on by a force 2Sxgp tending to
bring it back. The time of oscillation is therefore (§ {a)) 27r \l -
and is the same for all liquids whatever be their densities.
INDEX.
Aberration gives hodograph of Earth's
orbit 53
Abscissae 452
Absolute acceleration 64
— motion 63
— unit of force, Gauss's, 188; British 190
Acceleration, definition 34; uniform 32;
variable 33; average 33; angular 57;
composition and resolution 34, 37
— directed to a fixed centre 45
— in a fixed direction 44
— in logarithmic spiral with uniform
angular velocity about the pole 295
— in Simple Harmonic Motion 74
— in straight line, uniform 43
— in uniform circular motion 36, 39, 42
— of momentum 178
Accurate measurements, necessity for
352
Action, Least 279
— Maximum 317
— Minimum 3 1 1
— Stationary 281
— Varying 282
Aeolotropic substance, an 647
Alteration of latitude by hemispherical
hill, or cavity 496; by a crevasse 497
Ampere's Theory of Electrodynamics
336
Amplitude of S. H. M. 71
Angle between two lines, definition of
441 note
Angle of repose 473
Angle, solid 482 ; round a .point 483 ;
subtended at a point 485
Angular acceleration 57
Angular measure, standard of 357
Angular velocity 54; unit of 55; com-
position of 107, 108
Anticlastic surface 120
Approximate treatment of physical
questions 391
Arc, definition of 438; projection of
an 439
Area of an autotomic plane circuit 445
Argument of S. H. M. 71
Atmosphere Homogeneous 695 ; sec
Homogeneous
Attraction not modified by interposition
of other matter 474
— is normal to equipotential surfaces
506
— integral of normal, over a closed
surface 510
— direct analytical calculation of 494
— law of, when a uniform spherical
shell exerts no attraction on an in-
ternal point 541
■ — law of gravitation 475
— of gravitating, electric, or magnetic
masses 478
— variation of, in crossing an attracting
surface 495
— of a circular arc for a particle at its
centre 499
— of a right cone for a particle at its
vertex 494 {c)
— of a cylinder on a particle in its axis
494 {^)
■ — of a cylindrical distrihttion of matter
508
• — of a uniform circttlar disc on a par-
ticle in its axis 494 {a)
— of an infinite disc ^g^
— of t7vo equal uniform discs, one posi-
tive, other negative 494 {d)
— of an Ellipsoid 535, 537 ; of homo-
geneous ellipsoid 538 ; Maclaurin's
Theorem 539; Ivory's Theorem 540;
Duhamel's application of Ivory's The-
orem 54 1
— of an ellipsoidal shell 535 ; on an in-
ternal particle 536
— of a uniform limited straight line on
an external particle 499 {h)
— of a viouniain on a plumb line
496- (a)
— at the top and the bottom of a pit
496 {b)
— oi infinite parallel planes 508
— of a sphere composed of concentric
shells of uniform density 498
— of a uniform sphere on an external
particle infinitely near its surface 488
cor.
— of a uniform sphere 534, 541
INDEX.
387
Attraction of an uninsulated sphere
under the influence of an electrified
particle 493
— of a uniform spherical shell on an
internal point 479 ; converse proposi-
tion 541
— of a uniform spherical shell on an
external point 488
— oi a spherical surface whose density
varies as D~^ from exccTitric points
490 et seq.; excentric point inside at-
tracted point outside, and vice versa
491 ; excentric and attracted point
both within or both without 492
Autotomic circuit 443
Average curvature 14
Average stress 645
Average velocity 26
Axiom, physical 209 ; regarding the
equilibrium of a non-rigid body
584
Axis of a couple 201
Axis, central 579
Balance, Coulomb's Torsion 385
— requisites for a good 383
• — sensibility, stability and constancy
of a 384
— statical principles of 592
Balance, spring 386
Ballistic pendulum 263, 272
Bending of a supported beam or uniform
bar 625; supported at ends or middle
625; at ends and middle 627
Bending, effect of, on cross section of
body 679
Bifilar suspension 388
Body, motion of a rigid 106
Body, a perfectly rigid, defined 393,
401
Bramah Press, hydrostatic principle of
689
British system of units of mass 661 note
British absolute unit of force 1 90
Cardioid 105
Catenary 594 ; a parabola 599 ; kinetic
question relative to 598 ; inverse pro-
blem 600
Cathetometer 382
Central axis 579
Central ellipsoid 238
Centre of gravity, and centre of inertia
^95* 542, 582; centrobaric bodies
542 ; if it exist is centre of inertia 550 ;
position of in stable equilibrium 585,
in rocking stones 586; of a body in
equilibrium about an axis 587, on a
fixed surface 588; Pappus' theorem
concerning 589
Centre of pressure 688, 702
Centre of mass or inertia 195, 582 ;
motion of centre of inertia of a rigid
body 232, 550; moments of inertia
of centrobaric body round axes
through centre of inertia 551
Centrobaric body 542, proved possible
by Green 543, properties of 545 ;
centrobaric shell 547 ; centrobaric
solid 549 ; moments of inertia of a
centrobaric body round axes through
centre of inertia 551
Change of velocity 177, of momen-
tum 177
Characteristic function, Hamilton's 283
Chasles on confocal ellipsoids 537
Chronometer 367
Chronoscope 369
Circuit, linear 443 ; autotomic 443
Circular measure, unit of 357
Clairault's formula for the amount of
gravity at a place 187
Clocks 367
Closed curve 443
— polygon 443
Closed surface, ^7V(/(r, over a 510
Coarsegrainedness 646
Coefficient of elasticity 265 note, 644
Coefficient of restitution 265 ; of glass,
iron, wool 265
Comet, hodograph of orbit of 49
Component velocity 29 ; acceleration
37; of a force, effective 193
Composition of Velocities 31 ; Accelera-
tions 34 ; Simple Harmonic Motions in
same direction 75, in different direc-
tions 80; Angular velocities 107,
about axes meeting in a point 108 ;
Rotations 107, successive finite rota-
tions 109; Forces 221, of two acting
on a point 419, 422, special cases of
423 et seq.; nearly conspiring 427,
nearly opposed 428, at right angles
429, of any set of forces acting on a
rigid body 570 ; Couples in same plane
or in parallel planes 561, 562, 563, any
number, 564 ; not in parallel planes
565, any number of 566, and a force
568
Compound pendulum, Appendix g.
Compressibility 65 1
Conditions of equilibrium of a particle
408 ; a material point 470 ; of parallel
forces 558 ; of floating bodies 702—9 ;
of any number of couples 567 et seq.
Cone, orthogonal and oblique section of
very small 486 ; solid angle of 482;
area of segment cut from spherical
surface by a small cone 487
Cones opposite or vertical 481
288
INDEX,
Confocal ellipsoids, correspondingpoints
on 535; Chasles' proposition 537
Conical pendulum, Appendix/"
Conical surface 480
Conservation of energy 250
Conservative system 243
Constancy of a balance 384
Constraint of a point 165, of a body
167 ; one degree of constraint of the
most general character 170
Contrary forces 555 note
Continuity, equation of 16-2
Conversion of units : — pounds per sq.
inch to grammes per sq. centimetre
661 ; other units 362—366
Co-ordinates 452; propositions in co-
ordinate , geometry 459
Cord round cylinder 592, 603
Corresponding points in confocal ellip-
soids 535
Cosines, sum of the squares of the direc-
tion, of a line, equal to unity 460
Couple 201, axis of 201, moment of
201, direction of 560
— composition of in same or parallel
planes 561; any number 564; any
number not in parallel planes 566 ;
conditions of equilibrium of 567;
and a force, composition of 568 et seq.
Curvature of a plane curve 9 ; integral
14; average 14; of a surface 120; of
oblique sections, Meunier's Theorem,
121; principal, Euler's Theorem 122
Curvature of a lens, how to measure 381
Curve, plane 11; tortuous 13; of double
curvature i r ; continuous 35 ; closed
443
Curves use of, in representing experi-
mental results 347
Cycloid 6()^ 103; properties of 104;
prolate 103; curtate 103
D'Alembert's Principle 230
Day, Sidereal and Mean Solar 357
Degrees of freedom and constraint 165,
of a point 165, of a body 167; one
degree of freedom of most general
character 170
Density 1 74 ; linear, surface, volume,
477 ; mean density of the e&rth ex-
pressed in attraction units 715
Developable surface 125; practical con-
struction of a, from its edge 133
Diagonal scale 372
Direction of motion 8
Direction of rotation, positive 455
Direction cosine 463 ; sum of squares
of, equal to unity 4*60 ; of the common
perpendicular to two lines 464
Displacement of a plane figure in its
plane 91, examples 96; of a rigid
solid 100
Dissipation of Energy, instances 247,
292, 683
Dissipative systems 292
Distortion, places of maximum, in a
cylinder 678
Distribution of the weight of a solid on
points supporting it 626
Double-v^'eighing 384
Duhamel's application of Ivory's theo-
rem 541
Dynamics i
Dynamometer 389, Morin's 389
Edge of regression 132
Elastic body, a perfectly 643
— centre of a section of a wire 608;
line of elastic centres, 608; rotation
of a wire about 628
— curve transmitting force and couple
619, properties of 620; Kirchoffs ki-
netic comparison, common pendulum
and elastic curve 620
— solid equilibrium of 667
— wire or fibre 605
Elasticity, co-efficient of 265 note', of
volume 65 1 ; of figure ^i, i
Electric images 528; definition 530;
transformation by reciprocal radius
vectors 531; electric image of a
straight line, an angle, a circle, a
sphere, a plane 531 ; application to the
potential 532; of any distribution of
attracting matter on a spherical shell
533; uniform shell eccentrically re-
flected 533 ; uniform solid sphere
eccentricrtliy reflected 534
Elemeiits of a force 184
Ellipse, how to draw an 19
Ellipsoid, central 237
Ellipsoid, attraction of a, 535 ; corre-
sponding points on two 535 ; Ellips-
oidal shell defined 535 ; attractibri of
homogeneous ellipsoidal shell on in-
ternal point 536; Potential constant
inside 536; Chasles' Proposition con-
cerning 537 ; equipotential surfaces of
^ 537; Maclaurin's Theorem 539;
Ivory's 540; comparison of the po-
tentials of two 537
Ellipsoid, Strain 141; principal axes
of 142
Empirical formulae, use of 350
Energy, kinetic 179; kinetic energy of
a system 234; energy in abstract
dynamics 241, 251; foundation of
the theory of energy 244; potential
energy of a conservative system 245 ;
conservation of E. 250; inevitable loss
INDEX.
289
of energy of visible motion 247 ; po-
tential energy of a perfectly elastic
body strained 644 ; energy of a
strained isotropic substance 6^6
Epicycloid, integral curvature of 14,
motion in 105
Epoch in simple harmonic motion 71
Equation of continuity 162 ; integral and
differential 163
Equations of motion of any system 258
Equilibrium of z. particle, conditions of
408, 470, on smooth and rough curves
and surfaces 473; conditions of equi-
librium offerees acting at a point 470 ;
conditions of equilibrium oi three forces
acting at a point 584; graphic test
o{ forces in equilibrium 414; condi-
tions for stable equilibrium of a body
585, rocking stones 586, body move-
able about an axis 587, body on a
fixed surface 588; neutral, stable, and
unstable equilibrium, tested by the
principle of virtual velocities 256,
energy criterion of 257; conditions
of equilibrium of parallel forces 558;
conditions of equilibrium of forces
acting on a rigid body 576; equi-
librium of a non-rigid body not af-
fected by additional fixtures 584, of
a flexible and inextensible cord 594 ;
position of equilibrium of a flexible
string on a smooth surface 601, rough
surfaces 602 ; equilibrium of elastic
solid 667, of incompressible fluid com-
pletely filling rigid vessel 696, under
any system of forces 697 ; equilibrium
of a floating body 704 et seq., of a
revolving mass of fluid 710
Equipotential surfaces, examples of 499,
505, 526, of ellipsoidal shell 537
Equivalent of pounds per square inch
in grammes per square centimetre
661 ; other units 362 — 366.
Experience 320
Experiment and observation 324; rules
for the conduct of experiment 325 ; use
of empirical formulae in exhibiting
results of experiment 347
Euler's theorem on curvature 122, on
Impact 276
E volute 20, 22
Flexible and inextensible line, Kine-
matics of a t6; flexible and inexten-
sible surface, flexure of 125, general
property of 134; flexible string on
smooth surface, position of equili-
brium of 601, on rough surface 602
Flexure of flexible and inextensible sur-
face 125, of a wire 605 ; laws of flexure
and torsion 607 ; axes of pure flexure
609 ; case in which the elastic central
line is a normal axis of torsion 609 ;
where equal flexibility in all directions
610; wire strained to any given spiral
and twist 612; spiral spring 014;
principal axes of 679 ; distortion of the
cross section of a bent rod 679
Floating bodies, stable equilibrium of,
lemma 704 ; stability of 705 et seq. ;
see Fluid
Fluid, properties of perfect 401, 684;
fluid pressure 685, equal in all di-
rections 686, proved by energy cri-
terion 689 ; fluid pressure as depending
on external forces 690; surfaces of
equal pressure are perpendicular to
lines of force 691, are surfaces of
equal density and equal potential 692 ;
rate of increase of pressure 694, in
a calm atmosphere of uniform tem-
perature 695 (free surface in open
vessel is level 696) ; resultant pres-
sure on a plane area 702 ; moment of
pressure 702 ; loss of apparent weight
by immersion 703 ; conditions of equi-
librium of a fluid completely filling a
closed vessel 696, under non-con-
servative system of forces 697, im-
aginary example 699, actual case
701 ; equilibrium of a floating body,
lemma 704, stability 705, work done
in a displacement 705, metacentre,
condition of its existence 709; oblate
spheroid is a figure of equilibrium of
a rotating incompressible homoge-
neous fluid mass 711; relation be-
tween angular velocity of rotation
and density with given ellipticity 712 ;
table of eccentricities and correspond-
ing angular velocities and moments
of momentum for a liquid of the
earth's mean density 717; equilibrium
of rotating ellipsoid of three unequal
axes 719
Fluxion 28
Forbes's use of Viscous in connection
with glacier motion 683
Force, moment of 46, about a point
199, source of the idea of 173, de-
fined 183, specification of a 184,
measure of a 185, measurement of
224, by pendulum 387 ; fo7-ce of grav-
ity, Clairault's formula for 187, in
absolute units 187, average, in Britain
191; unit of force, gravitation 185,
absolute 188; British absolute imit
191; attraction unit of force 476;
representation of forces by lines 192 ;
component of force 193 ; composition
19
290
INDEX.
of forces ii\, 418, parallelogram and
polygon of -2 22, true polygon of 416,
triangle of 410; forces through one
point, resultant of 412 ; forces in equi-
librium, graphic test of 414; resultant
of three forces acting at a point 465 ;
resolution of force along three speci-
fied lines 468 ; resultant of any number
of forces acting at a point 469 ; con-
ditions of equilibrium of forces acting
at a point 470; resultant of forces
whose lines meet 552 ; two parallel
forces in a plane 554, in dissimilar
directions 555, of any number 556,
not in one plane 557, conditions of
equilibrium of 558 et seq. ; forces and
a couple 568 ; forces may be reduced
to one force and one couple 570 ; re-
duction of forces to simplest system
571; parallel forces whose algebraic
sum is zero exert a directive action
only 583; conditions of equilibrium
of three, acting on a rigid body 584 ;
conservative system of 243
Force in terms of the potential 504 ; at
any point, due to attraction of a
spherical distribution of matter, a
cylindrical distribution, or a distri-
bution in infinite planes 508, where
it varies as Z>~^ 509
Force of gravity, Clairault's formula for
187, in absolute units 187, average
value in Britain r9i, in Edinburgh
191, law of 475
Force, line of, definition 507, instance
of 499 ; variation of intensity along
a 508
Force, tube of 508
Form of equilibrium of a rotating mass
of fluid 711 — 719
Formulae, use of empirical, in exhibit-
ing results of experiment 347
Foucault's pendulum 87
Fourier's theorem 88
Freedom of a point, degrees of 165; of
a rigid system 167
Friction brake, White's 390
Friction, laws of statical 403, kinetic
404 ; effect of tidal friction 248 ; fric-
tion of liquids varies as the velocity
292 ; friction of solids 293 ; of a cord
round a cylinder 592, 603
Gauss's absolute unit of force i88j
theorem relating to potential 5r5
Geodetic line 124, properties of 601
Glacier Motion, Forbes's Viscous The-
ory of, meaning of Viscous in 683
Gravitation, law of 475 ; potential 50:5
Gravity, force of, Clairault's formu'la
for 187, at Edinburgh, in Britain
191, in absolute units 187, work
done against 509
Gravity, centre of, and centre of inertia
I95> 582 ; centre of gravity 542 ; pro-
perties of a body possessing a centre of
gravity 544 ; centre of gravity where
it exists coincides with centre of inertia
550; position of centre of gravity in
a body for stable equilibrium 585, in
rocking stones 586, in a body with
one point fixed: with two points
fixed 587, on a surface 588; Pappus'
theorem concerning, sometimes called
Guldinus' theorem 589
Green 501; problem in potential 517;
the general problem of electric in-
fluence possible and determinate 521
Gyration, radius of 235
Gyroscopes, motion of 116 .
Hamilton's Characteristic Function 283
Harmonic motion 69 ; simple harmonic
motion 70, amplitude, argument,
epoch, period, phase, 71, instances of
72; velocity in simple harmonic mo-
tion 73, cuceleration in 74 ; composi-
tion of two simple harmonic motions
in one line 75, examples 77 ; graphical
representation of simple harmonic
motion in one line 79; cotnposition
of simple harmonic motion in different
directions 80; of different kinds in
different directions 84; in two rect-
angular directions 85
Harton coal mine experiment 498
Hodograph 49, of a planet or comet
49, of a projectile 50, of motion in
a conic section 51, of path where
acceleration is directed to a fixed
point and varies as D'"^ 6r
Homogeneous atmosphere defined 695 ;
height of 694, at Paris, at Edinburgh
695.
Homogeneous body 646
Homogeneous strain 135; see Strain
Horsepower 240
Hydrodynamics 683 ; see Fluid
Hydrostatics 685 ; see Fluid
Hyperbola, how to draw a 19
Hypocycloid and hypotrochoid 105
Hypothesis, use of 332
Image, electric 528; see Electric im-
ages
Impact 259, duration of 259 ; time in-
tegral 262 ; ballistic pendulum 263 ;
direct impact of spheres, Newton's
experiments on 265, loss of kinetic
energy in 266, due to 267, case with
INDEX,
291
no loss of kinetic energy 268 ; mo-
ment of an impact 272; work done
by 273; Euler's theorem 276
Impressed force 183; see Force
Inclination of two given lines in terms
of their direction cosines 464
Inertia 182
Inertia, centre of 195; see Centre of
Mass
Inertia, moment of 198, 235 — 239, of
a centrobaric body 551
Inextensible line 16, surface 125, gen-
eral property of 134
Instability of motion 300; instances
302, 303, 304
Interpolation in physical experiments
350
Involute 20
Isotropic substance 647
Isotropy, conditions fulfilled in elastic
650, in one quality and aeolotropy
in others 648
Ivory's theorem on homogeneous con-
focal ellipsoids 540
Kepler's first law a consequence of ac-
celeration directed to a fixed point 45
Kilogramme 365
Kinematics 4, of a point 7, of an in-
extensible and flexible line 16, of a
plane figure 91, flexible and inexten-
sible surface 127
Kinetic energy 179, rate of change of
180; gain in kinetic energy equiva-
lent to work done 207 ; kinetic energy
of a system 234; loss of kinetic en-
ergy in direct impact 266
Kinetic foci 310 — 319, number of, in
any case 316
Kinetic friction 404
Kinetics 2, 3, 4
Kinetic stability 300; kinetic stability
or instability discriminated 301 ; cases
of kinetic stability 302, 303, 304;
kinetic stability in a circular orbit
304; oscillatory kinetic stability 308;
general criterion of kinetic stability
309 ; motion on anticlastic surfaces is
unstable, synclastic stable 309
Kinetic symmetry 239
Kirchoff's kinetic comparison between
twisting a wire and the motion of a
pendulum 620
Latitude altered by attraction of a
mountain, or hemispherical hill, or
cavity 496, by a crevasse 497
Laws of energy, dynamical, 252
Laws of friction 403 ; see Friction
Laws of motion, history of 208 j first
law 210, second 217, third 227,
Scholium 229, 241
Least action 2 79
Least squares, method of 340
Lemniscate, integral curvature of 14
Lengthening of a spiral spring due to
torsion 618
Level surface 505
Limitation of dynamical problems 391
Line density 477
Line, expression for a, in co-ordinates
.459
Line of elastic centres remains un-
changed in length 608 ; see Elastic
Line of force def. 507, instances of 499,
variation of intensity along a 508
Line, orthogonal projection of a 442
Liquid, effective moment of inertia of
675, note
Locus of centre of curvature 22
Longitudinal vibrations, velocity of
transmission of 658
Longitudinal rigidity 657
Loss of weight of body immersed in
fluid 703; see Fhdd
Lunar tides 77
Machines, science of i
Maclaurin's theorem on homogeneous
confocal ellipsoids 539
Mass 174; measurement of 175, 224,
unit of 190, 365, 476, 715, British
unit of 190; mass v. weight 17 = ,
186
— centre of 196; see Centre of Mass
Matter 173
Maximum action 317
Mean angular velocity 58
Mean density 174; of Earth, Sche-
hallien experiment 496, Harton coal
mine experiment 498
Mean solar day 357
Measure of time 358, 371, of length
360, of surface 363, of volume 364,
of mass 365, of force 366, of work
366, of angles 357, of pressure 661
Measurement of force 185, 224, of
masses 224
Mechanical powers 591; balance 592
Mechanics i
Mechanism 4
Metacentre 709 ; conditions for its ex-
istence ; see Fluid
Method of least squares 340
Method of representing experimental
results 347
Metre 362
Meunier's theorem on curvature 121
Micrometer 379
Minimum action 311; two or more
292
INDEX.
courses of minimum action possible
Modulus of elasticity, Young's 657 ;
weight and length of modulus 658;
specific modulus of isotropic body
Moment about a pomt, of a velocity or
a force 46, representation of 199, of
a couple 201, of an impact 272, of
pressure 702
Moment of inertia 198, 235, of a cen-
trobaric body 551
Moment of momentum 202, of a rigid
body 232
Momentum 176, change of 177, accele-
ration of 178
Motion of a material particle 7; rela-
tive motion ()i\ simple harmonic mo-
tion 69; of troops on suspension
bridge 78 ; of point of vibrating string
79 ; of a plane figure in its own plane
19 ; of a rigid body about a fixed point
106 ; general motion of a rigid body
112; of a screw in its nut 113;
quantity of motion 176; Newton's
laws of motion 208, see Laws ; re-
sistances to motion 247 ; motion in
a resisting medium 292, in a logarith-
mic spiral 295 ; of a system slightly
disturbed from a position of equilib-
rium 290
Neap tides 77
Neutral equilibrium 256; of floating
bodies 705
Newton's laws of motion 208, seeZazfj;
experiments on impact 265
Non-conservative system 298
Normal 22
Normal attraction over a closed surface,
integral of 510
Oblique coordinates 453
Observation and experiment 320
Opposite cones 481
Opposite forces 5^5, note
Ordinates 452
Orthogonal projection 442
Oscillation in U tube, Appendix k
Parallel forces in a plane, resultant of
two 554, in dissimilar directions 555,
of any number 556, not in one plane
557, equilibrium of 558
Parallelogram of velocities 31, of forces
219
Particle material v. geometrical point
7, 181
Pendulum, Robins' ballistic 263, 272 ;
pendulum as a measurer of force 387 ;
simple pendulum Appendix (a) ; com-
pound pendulum Appendix \g)
Perfect fluid 401, 684
Perfect solid, ideal 656
Perfectly elastic body 643; potential
energy of perfectly elastic body held
strained 644
Period of simple harmonic motion 7 1
Periodic disturbance 306
Periodic function, Fourier's theprem
regarding 88
Perpetual motion the, is impossible 244
Phase of simple harmonic motion 71
Physical axiom 209; concerning equi-
librium 584
Plane, osculating 1 2 ; motion of plane
figure in its own plane 91
Planet, path of 45 ; hodograph of 49, 51,
61
Plasticity 683
Polar coordinates 457, 459
Polygon plane ii, gauche 11, closed
443, of velocities 31, of forces 219
Potential 500; the mutual potential
energy of two bodies 502, at a point
503, force in terms of potential 504 ;
equipotential surface 505 ; potential
due to an attracting particle 509, to
any mass 509, potential cannot have
a maximum or minimum value at a
point in free space 511, cases of this
515, 516; has same value throughout
the interior as at the surface of a closed
space 513; mean value of potential
throughout a sphere equal to the value
at centre 514; Gauss's Theorem 515;
Green's problem 517; potential due
to a uniform spherical shell 514) 533 ;
how to distribute matter so as to get
a given potential 517 — 521 ; potential
due to uniform sphere 534; due to
ellipsoidal shell 536
Potential energy due to work done 207,
of a conservative system 245 ; the
mutual potential energy of two bodies
502, of elastic solid held strained 644
Precession 117
Precessional rotation 116
Pressure, centre of 688, 702 ; pressure
at a point in a fluid same in every
direction 685, 687, 689; surfaces of
equal pressure are level surfaces 691 j
whole pressure 709
Principal axes of a strain 144; st&Sirain
Principal axes of inertia 237
Probable error 343
Probable result from a number of obser-
vations, deduction of the 338; method
of least squares 340 ; practical appli-
cation 345
INDEX.
293
Projectile, path of 44; hodograph 50
Projection of areas 200 ; of a point on a
straight line 437; orthogonal projec-
tion 442, of a shell 447, of any two
shells, of a closed surface 448, of
equal areas in parallel planes 449, of
a plane figure 449
Pulley, kinematics of 18
Pure strain 159; see Strain
Radius of curvature 9
Radius of g>Tation 235
Regression, edge of 132
Relative motion 63 et seq. ; acceleration
of 64
Repose, angle of 404, 473
Residual phenomena 328
Resistance to motion 247, 250; varying
as the velocity in fluids 293; to
change of shape, frictional 683
Resisting medium 247
Resolution of velocity 30, of forces 431,
geometrical solution 432, trigonomet-
rical solution 433, in directions at
right angles 434 ; application to find
the resultant of a number of forces
acting on a point 435 ; resolution of
forces alng three specified lines 468
Rest 211
Restitution, co-efficient of 265
Resultant velocity 31 ; resultant of forces
on a point 412, 419; three forces act-
ing on a point 465, any number 470
Revolving mass of fluid, equilibrium of
710; see Fluid
Rigid body, displacement of 90, motion
of 106, general motion of 112, rigid
body defined 393, 401
Rigidity 651; longitudinal 65 7: rigidity
and resistance to compression 655;
rigidity as depending on form 677
Rocking stones 586
Rolling of bodies 109; of curve upon
curve 100
Rolling motion 118, 119
Rope round cylinder 592, 603
Rotation, positive direction of 455
Rotations about parallel axes, compo-
sition of 98 ; composition of rotation
and translation in one plane 99 ; ro-
tations of a rigid body, composition of
106; successive finite rotations 109
Rotation of a wire round its elastic
central line 628 ; see Elastic
Schehallien experiment 496
Scholium to law ill 229, 241
Screw, motion of a, in its nut 113,
337
Sea mile 361
i4»
Section of a small cone, oblique 486
Sensibility and stability of a balance 384,
592
Shape, change of, involves dissipation
of energy 683, 247
Shear, simple 150, axes of a 152,
measure of a 153, combined with a
simple elongation and expansion 156
Shell def 446
Siderial day 358
Simple linear circuit 443
Simple harmonic motion 70, in me-
chanism 72, composition of, in one
line 75, examples 77, composition of,
in different directions 80, of diflerent
kinds in different directions 84, in two
rectangular directions 85 ; see Har^
monic
Simple pendulum, Appendix {a)
Simple shear 150 ; axes of a shear 152 ;
ratio of a shear 153, amount of a
154; planes of no distortion in a 155
Solar system, ultimate tendency of the
249
Solar tides 77
Sohd angle 482; round a point 483;
subtended at a point 485
Solid, elastic 643, 651; potential energy
of elastic solid held strained 644 ; fun-
damental problems of the mathemati-
cal theory of the equilibrium of an
elastic solid 667; equations of internal
equilibrium of 668; St Venant's ap-
plication to torsion problems 669 ;
small bodies stronger than large ones
in proportion to their weight 682 ;
imperfectness of elasticity in solids
683
Space described under uniform accele-
ration in direction of motion 43
Space, British unit of 190
Specific modulus of elasticity 689
Sphere, attraction of, composed of con-
centric shells of uniform density 498 ;
attraction of uniform sphere and po-
tential due to 634 ; see Attraction and
Potential
Spherical shell, uniform, attraction on
internal point 479, external point
488, on an element of the surface
489, potential due to 533; see At-
traction and Potential
Spherometer 380
Spinning motion 118
Spiral, motion in logarithmic 295, 296
Spiral springs 386, as measurers of force
386,614, kinetic energy of 616, length-
ening of, due to torsion 6i8
Spring balance; see Spiral springs
Spring tides 77
294
INDEX.
Stable equilibrium 256, 257; see Centre
of Gravity and Floating Bodies
Stability of motion 300
Static friction 404
Statical problems, examples of 591;
balance 592; rod with frictionless
constraint 592 ; rod constrained by
rough surfaces 592 ; block on rough
plane 592 ; mass supported by rings
round rough post 592; cord wound
round cylinder 592
Statics 2, 3, of a particle 408, of a
rigid body, 552
Stationary action 281
Straight beam infinitely little bent
623
Strain 135 ; homogeneous strain 136 ;
properties of homogeneous strain 137 ;
strain ellipsoid 141 ; axes of strain
ellipsoid 144 ; elongation and change
of direction of any line of a body in
condition of strain 145 ; distortion
in parallel planes without change of
volume 148 ; simple shear 150 ; axes
of a shear 152, ratio of a 153, amount
of a 154, planes of no distortion in
a 155, is a simple elongation and ex-
pansion combined with a shear 156;
analysis of strain 157 ; pure strain
159; composition of pure 160
Stress 629, homogeneous 630, specifi-
cation of a 632, components of a
633, simple longitudinal and shearing
stress 633 ; stress quadratic 634 ;
normal planes and axes of a stress
quadratic 635 ; varieties of stress
quadratic 636 ; laws of strain and
stress compared 639; rectangular ele-
ments of strain and stress 640 ; work
done by a strain 641 ; a physical ap-
plication 642 ; stress produced by a
single longitudinal stress 653 ; ratio
of lateral contraction to longitudinal
extension different for different sub-
stances 655 ; stress required to pro-
duce a simple longitudinal strain 663 ;
stress components in terms of strain
for isotropic body 664 ; strain compo-
nents in terms of stress 665 ; funda-
mental problems in mathematical
theory of equilibrium of elastic solid
667 ; equation of energy of isotropic
body 666 ; equations of internal
equilibrium 668 ; comparative strain
of similar bodies as depending on
dimensions 682
St Venant on torsion of prisms 669 ;
see Torsion
Surface density 477
Surface of equilibrium ^05 ; relative in-
tensity of force at different points of
a 506
Surfaces of equal pressure in a fluid at
rest are also surfaces of equal potential
and equal density 692
Symmetry, kinetic 239
Symmetrical co-ordinates 458, 459
Synclastic surface 120
System, conservative 243
— non-conservative 298
Tidal friction 247, effect of Tides in
lengthening the period of the Earth's
rotation about her axis 248, 249
Tides 77
Time, unit of 190, measurement of 213,
.358, 371
Time integral 262
Time of rotation of the earth round its
axis increased by friction 248
Time of oscillation of fluid in a U tube
Appendix k', of a simple pendulum
Appendix b, c, d, e, compound pen-
dulum Appendix g; wave running
along a stretched cord Appendix h
Tops, motion of spinning 118
Torsion, laws of 607
Torsion balance 383
Torsion of a wire 605 ; laws of 607
Torsion of prisms, St Venant on 669,
lemma 670; torsion of circular cy-
linder 671 ; prism of any shape 672,
623 ; hydrokinetic analogue 675 ; con-
tour lines for normal sections of
prisms &c. under torsion ; elliptic
cylinder, equilateral triangular prism,
curvilinear square prisms, square
prisms: bars elliptic, square, flat,
rectangular 676 ; relation of tor-
sional rigidity to flexural rigidity
677; ratio of torsional rigidity to
those of circular rods of same mo-
ment of inertia, or of sariie quantity
of material 677; places of greatest
distortion in twisted prisms 678
Tortuosity 11
Tortuous curve I r, 13
Transformation electrical, by reciprocal
radius vectors 531
Transmission of force through elastic
solid 629 ; transmission of homo-
geneous stress 630 ; force trans-
mitted across any surface in elastic
solid 631
Triangle of forces 410, equivalent to a
couple 411
Triangle of velocities 3 1
Trochoid 103
Tubes of force 508
Turning, positive direction of 455
INDEX.
295
Uniform acceleration 36, 43; space
described 43
Uniform circular motion, acceleration
in 37; composition of two 86
Uniform velocity 23
Unit angle n ; of angular velocity 55 ;
— of angular measure 357
— of cubic measure 364
— of force 188, 366, 476
— length 360 — 362
— mass, space, time 190, 565
— work (s9ientific) 204, gravitation 204
— surface 363
Units, tables for conversion of 362 — 366,
66r
Unstable equilibrium 256, 257; see
Centre of Gravity and Fluid
Varying action 2 79 ; optical illustration
286 ; a criterion for kinetic stability
309
Velocities, parallelogram, triangle, poly-
gon of 31 ; examples of velocities 41
Velocity 23. uniform 23, variable 26,
component 29, resolution of 29,
resultant 31, moment of 46, angular
54, relative d^, change of 177,
virtual 203, 254
Velocity of a planet at any point of its
orbit 48 ; in simple harmonic motion
Velocity of escape of fluid from an ori-
fice Appendix g
— of longitudinal vibrations along a
rod 658
— of wave along stretched cord Ap-
pendix h
Venant (St) on torsion 669; see Tor-
sion
Vernier 373
\'ertical cones 481
Vibrations produced by impact 220,
269; in a resisting medium 293;
along stretched cord Appendix h,
velocity of transmission of, through a
rod 658
Virtual velocity 203, 254, moment of
.203
Viscosity of solids 683 ; of fluids 683
Vis viva 179
Volume, change of involves dissipation
of energy 683
Volume, density 477, 715
\'olume, elasticity of 651
Weber's electrical theory 336
Weight V. mass 175 ; a measure of mass
175, 186
White's friction brake 390
Whole pressure on a submerged surface
702
Wire, flexure of a 622 ; see Flexure
Work 204, unit of 204, against force
varying inversely as square of distance
509, independent of path pursued
under conservative system of force
509, done in straining a perfectly
elastic body 644 ; transformations of
work 207
Yard 360
Young's modulus 657
CAMBEIOGK: FRINTEU by C. J. clay, M.A., AT THE UNIVERSITY PRF.SS.
UNIVERSITY OF CALIFORNIA LIBRARY
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APR 1 3 1856 It
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JUL 5 1962
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THE UNIVERSITY OF CALIFORNIA LIBRARY